p-Fractals and Hilbert-Kunz Series
A Dissertation
Presented to
The Faculty of the Graduate School of Arts and Sciences
Brandeis University
Department of Mathematics
Professor Paul Monsky, Advisor
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
Pedro Teixeira
May 2002
i
This dissertation, directed and approved by Pedro Teixeira’s Committee,
has been accepted and approved by the Faculty of Brandeis University in
partial fulfillment of the requirements for the degree of:
DOCTOR OF PHILOSOPHY
---------------------------------------------------------------------------------Dean of Arts and Sciences
Dissertation Committee:
---------------------------------------------------------------------------------Paul Monsky, Mathematics
---------------------------------------------------------------------------------Ira Gessel, Mathematics
---------------------------------------------------------------------------------Ragnar Buchweitz, Mathematics
University of Toronto
ii
Acknowledgments
I’d like to express my deepest gratitude to my advisor, Professor Paul
Monsky. If it weren’t for his insight, which he was always willing to share
with me, and his patience during moments of confusion, this work wouldn’t
have been possible. It is wonderful to work with a person with such a vast
knowledge, and who is never afraid to face the most difficult problems!
I would like to thank Professor Susan Parker, for helping me become a
better teacher and for giving me confidence when I needed it. Her enthusiasm
about teaching seems to be contagious, and during these years at Brandeis I
couldn’t help getting infected. :-)
I also wish to thank the Mathematics Department faculty and staff, and
my fellow students, for making my stay at Brandeis a pleasure.
Finally, I’d like to thank CAPES (Brası́lia-DF, Brazil) for the financial
support during my first four years.
iii
Abstract
p-Fractals and Hilbert-Kunz Series
A dissertation presented to the Faculty of the
Graduate School of Arts and Sciences of Brandeis
University, Waltham, Massachusetts
by Pedro Teixeira
In this dissertation we prove the rationality of the Hilbert-Kunz series of a
X
large family of polynomials, including polynomials of the type
Gi (xi , yi ),
i
where the Gi are homogeneous polynomials with coefficients in a finite field.
The proof requires the use of certain functions which show remarkable selfsimilarity properties. This behavior was formalized through the introduction
of the concept of p-fractals. Most of this work is devoted to the study of
the properties of p-fractals, and the proof that those functions which arise
in the study of certain Hilbert-Kunz functions are in fact p-fractals. Then
some formalism involving the representation ring Γ, introduced by Han and
Monsky ([HM93]), gives the rationality results.
We conclude by analyzing a family of four dimensional fractals, and we
obtain some classification results that may be used in the explicit calculation
of Hilbert-Kunz series.
iv
Contents
1 Preliminaries
1.1
1.2
7
The Representation Ring Γ . . . . . . . . . . . . . . . . . . . .
7
1.1.1
Some Product Formulas in Γ. . . . . . . . . . . . . . . 10
1.1.2
The Function φ. . . . . . . . . . . . . . . . . . . . . . . 11
Regular Sequences . . . . . . . . . . . . . . . . . . . . . . . . 15
2 p-Fractals and the Main Theorem
23
2.1
p-Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2
Syzygy Gaps and δ . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3
The Functions δH and ϕH . . . . . . . . . . . . . . . . . . . . 37
2.4
The Function δa . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5
The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6
Other p-Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Four Dimensional Fractals
3.1
61
Ideal Classes and the Fractals δa . . . . . . . . . . . . . . . . . 62
v
3.1.1
Ideal Classes with δa (0) ≥ 4 . . . . . . . . . . . . . . . 62
3.1.2
Ideal Classes with δa (0) = 3 . . . . . . . . . . . . . . . 62
3.1.3
Ideal Classes with δa (0) = 2 . . . . . . . . . . . . . . . 63
3.1.4
Ideal Classes with δa (0) = 0 . . . . . . . . . . . . . . . 64
3.1.5
Interlude: Reflections . . . . . . . . . . . . . . . . . . . 74
3.1.6
Back to Four Dimensions . . . . . . . . . . . . . . . . . 81
3.1.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2
Symmetry Theorems . . . . . . . . . . . . . . . . . . . . . . . 83
3.3
Blow-Up Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 86
vi
Introduction
The Hilbert-Kunz function of a Noetherian ring A of characteristic p > 0
with respect to an m-primary ideal q (where m is a maximal ideal) is the
function
n
n 7→ en (A, q) := length(A/q[p ] ),
n
n
where q[p ] := (xp | x ∈ q) is the pn -th Frobenius power of q. So it resembles
the Hilbert-Samuel function, with powers replaced by Frobenius powers. It
is closely connected to the theory of tight closure. Suppose the height of the
maximal ideal m is d. In [Mon83] Monsky showed that en (A, q) = cpdn + ∆n ,
where c is a positive real constant, known as the Hilbert-Kunz multiplicity of
A with respect to q, and ∆n = O(p(d−1)n ). In the same paper, it was also
shown that when d = 1 the constant c is the Hilbert multiplicity of the ring
A with respect to q, and ∆n is periodic for n 0. However, when d ≥ 2,
except for Kunz’s result that c ≥ 1, nothing else is known about c and ∆n ,
and the calculation of en (A, q) is often a challenge.
If g ∈ k[x1 , . . . , xs ], where k is a field of characteristic p, the Hilbert-
1
Kunz function of k[x1 , . . . , xs ]/(g) with respect to the maximal ideal m :=
(x1 , . . . , xs ) is usually referred to simply as the Hilbert-Kunz function of g,
and denoted by en (g). So
n
n
n
n
en (g) = deg(xp1 , . . . , xps , g) = dimk k[x1 , . . . , xs ]/(xp1 , . . . , xps , g).
The Hilbert-Kunz series of the polynomial g is the power series
∞
X
en (g)tn .
n=0
One suspects it is always a quotient of two polynomials, but there is no
effective way of computing it in general.
In [HM93] Han and Monsky computed the Hilbert-Kunz function of polynomials of the type xd11 +· · ·+xds s . In particular, their results implied that the
Hilbert-Kunz series of such polynomials are rational. The main idea behind
those calculations was the introduction of the representation ring Γ—whose
basic properties will be reviewed in chapter 1—, and the determination of
some multiplication formulas in Γ.
In [CH98], Chiang and Hung determined the Hilbert-Kunz functions of
polynomials that can be written as a sum of monomials in disjoint sets of
variables (e.g. xl11 · · · xlrr + y1m1 · · · ysms + z1n1 · · · ztnt ), by introducing an alternative product in the representation ring.
The main result of this dissertation is the proof of rationality of the
Hilbert-Kunz series of a large family of polynomials—including all polyX
nomials of the type
Gi (xi , yi ), where the Gi are homogeneous, and the
i
field k is finite. For example, if p = 5, then the Hilbert-Kunz series of
2
g := x41 + y14 + x2 y2 (x22 + x2 y2 + y22 ) is
1 + 199t − 145t2 − 727t3
.
(1 − 125t)(1 − 3t)(1 − 2t − t2 )
√
√
√ n
√ n
129+67 √2
129−67 √2
164 n
3n
So en (g) = 1260912
5
−
3
+
(1
−
2)
+
(1
+
2) .
468907
61
250+126 2
250−126 2
The proofs require the use of certain functions which show striking selfsimilarity properties. We found that the correct way to formalize this behavior was through the introduction of the concept of p-fractals, which will be
studied in detail in chapter 2.
If H ∈ k[x, y] is a homogeneous polynomial, then there is a (unique)
continuous function ϕH : [0, 1] → [0, ∞) whose value at aq (q a power of p, a
an integer with 0 ≤ a ≤ q) is:
ϕH
a
q
:=
1
· deg(xq , y q , H a ).
2
q
The key to the results stated above is the proof that ϕH is a p-fractal, for
any homogeneous polynomial H ∈ k[x, y], when the field k is finite. This,
together with some formalism involving the representation ring—which will
be covered in chapter 1—, gives the desired rationality theorem.
We’ll show that ϕH is a p-fractal by relating it to another function δH .
Given nonzero forms F , G and H in k[x, y], we define δ(F, G, H) := deg F +
deg G + deg H − 2m, where m is the smallest degree of a nontrivial relation
between F , G and H. If F , G and H have no common factor then δ(F, G, H)
is related to the degree of the ideal (F, G, H) by the formula:
1
δ(F, G, H)2
deg(F, G, H) = (2d1 d2 + 2d1 d3 + 2d2 d3 − d1 2 − d2 2 − d3 2 ) +
,
4
4
3
where d1 , d2 and d3 are the degrees of F , G and H. Now δH is the continuous
function [0, 1] → R whose value at aq (0 ≤ a ≤ q = pn ) is
1
q
· δ(xq , y q , H a ).
From the connection between deg(F, G, H) and δ(F, G, H) mentioned above,
the desired result reduces to showing that δH is a p-fractal.
The proof that δH is a p-fractal involves passing from one to many dimensions, introducing the continuous functions δa : [0, 1]r → R whose values
r
Y
ai
a1
ar
n
q
q
at ( q , . . . , q ) (0 ≤ ai ≤ q = p ) are δ F , G , `i , where F and G are
i=1
homogeneous polynomials and the `i are pairwise prime linear forms, and F ,
Q
G and `i have no common factor. We’ll show that all such functions are
p-fractals when the field k is finite.
In chapter 3 we’ll focus on a family of four dimensional fractals. We’ll
prove some classification theorems that allow us to find certain “blow-up
rules”, which completely determine those fractals. These blow-up rules may
be used in the explicit calculation of certain Hilbert-Kunz series. We’ll also
show that those fractals have some interesting symmetry properties.
4
Notation
Throughout this dissertation, p will always be a prime number, q a power of
p, and k a field of characteristic p.
The degree k component of any graded ring or module M will be denoted
by Mk . M (d) is the module or ring obtained from M by shifting the degrees
by d; namely M (d)k := Md+k .
If a is a zero dimensional ideal of a k-algebra A, then deg a denotes the
degree of a, i.e. dimk A/a.
If S is any statement which can be true or false, then the bracketed
notation [S] stands for 1 if S is true, 0 otherwise. For instance, [p|n] is 1 if
p divides n, otherwise 0.
If n is a nonnegative integer, the set {0, 1, . . . , n} will be denoted by [n].
Notation connected with the representation ring, Γ, will be introduced in
chapter 1; notation connected with syzygies and p-fractals in chapter 2.
5
Other notation used throughout is:
en (A, q)
Hilbert-Kunz function of the ring A with respect to the
m-primary ideal q (m maximal)
a[q]
en (g)
q th Frobenius power of a, a[q] := (xq | x ∈ a)
Hilbert-Kunz function of k[x1 , . . . , xs ]/(g) with respect
to the ideal (x1 , . . . , xs )
6
Chapter 1
Preliminaries
1.1
The Representation Ring Γ
In this section we’ll present a brief account on the representation ring Γ—an
idea introduced by Han and Monsky in [HM93]. We’ll summarize some of
the formulas they found, and introduce an endomorphism φ : Γ → Γ.
Throughout this dissertation, p will always be a prime number, q will be
a power of p, and k will be a field of characteristic p. A k-object is a finitely
generated k[T ]-module on which T acts nilpotently. Γ is the Grothendieck
group of the semigroup of isomorphism classes of k-objects under the usual
direct sum. We introduce a product on Γ, which will give it a ring structure.
If η, ζ ∈ Γ are represented by the k-objects M and N , then ηζ is the image
in Γ of the k-object M ⊗k N , where T acts as follows: T (m ⊗ n) = (T m) ⊗
n + m ⊗ (T n). Now Γ endowed with this addition and multiplication is a
7
commutative ring, called the representation ring. The zero and identity
elements of Γ are respectively the images of the zero module and k[T ]/(T )
in Γ.
For any nonnegative integer n, let δn be the image of Mn := k[T ]/(T n )
in Γ. Note that in particular δ0 = 0 and δ1 = 1. The theory of modules over
PIDs shows that (Γ, ⊕) is a free abelian group, with basis {δ1 , δ2 , . . .}. We
construct a second basis {λ0 , λ1 , . . .}, where λn := (−1)n (δn+1 − δn ).
X
Note that δn =
(−1)i λi . So the λi -coordinate of δn , denoted [λi ]δn , is
i<n
(−1)i [i < n], which is the same as (−1)i dimk T i Mn /T i+1 Mn . In general, if
ω is the image of the k-object M in Γ then we can write ω = δn1 + . . . + δnk ,
and we see that
[λi ]ω = (−1)i dimk
T iM
M
M i
=
(−1)
dim
−
dim
.
k
k
T i+1 M
T i+1 M
T iM
Definition 1.1. Let g ∈ k[x1 , . . . , xs ]. Then hgin is the image in Γ of the
n
n
k-object k[x1 , . . . , xs ]/(xp1 , . . . , xps ), where T operates by multiplication by g.
In terms of the λi ,
hgin =
X
(deg(xq1 , . . . , xqs , g i+1 ) − deg(xq1 , . . . , xqs , g i ))(−1)i λi ,
i
where q := pn . However, the general term of this sum vanishes if i ≥ q. For,
if this is the case, then (xq1 , . . . , xqs , g i+1 ) = (xq1 , . . . , xqs , g i ) = (xq1 , . . . , xqs ). So
hgin =
q−1
X
(deg(xq1 , . . . , xqs , g i+1 ) − deg(xq1 , . . . , xqs , g i ))(−1)i λi .
i=0
8
Example 1.2. Let q := pn .
• hxin = δq (here we take s = 1).
• Let M := k[x, y]/(xq , y q ), where T operates by multiplication by xy.
Then dimk M/T i M = deg(xq , y q , xi y i ) = q 2 − (q − i)2 [i < q]. Thus
dimk M/T i+1 M − dimk M/T i M = (2q − 2i − 1)[i < q], and [λi ]hxyin =
(−1)i (2q − 2i − 1)[i < q]. Therefore
hxyin =
q−1
X
(2q − 2i − 1)(−1)i λi .
i=0
(Here we take s = 2.)
• Now suppose M := k[x]/(xq ) and T acts by multiplication by x2 . Then
dimk M/T i M = min(2i, q). Thus if p is odd, then
q−3
hx2 in = (−1)
q−1
2
λ q−1 +
2
X
2
2(−1)i λi = δ q−1 + δ q+1 ,
2
2
i=0
while if p = 2 we have hx2 in = 2δ 2q .
Definition 1.3. α : Γ → Z is the function u 7→ [λ0 ]u.
If g ∈ k[x1 , . . . , xs ], then
n
n
α(hgin ) = deg(xp1 , . . . , xps , g) = en (g).
If h is another polynomial in different variables, say h ∈ k[y1 , . . . , yt ], then
hgin hhin = hg + hin . This, together with the fact that α(λi λj ) = δij ([HM93,
Theorem 1.10]), allows us to compute a number of Hilbert-Kunz series of
9
polynomials that can be written as a sum of polynomials in two disjoint sets
of variables.
Example 1.4. We illustrate these ideas with a rather simple example, calculating the Hilbert-Kunz series of xy + zt ∈ k[x, y, z, t]:
en (xy + zt) = α(hxy + ztin )
= α(hxyin hxyin )
q−1
X
=
(2q − 2i − 1)2
i=0
q(4q 2 − 1)
=
3
So en (xy + zt) = 34 p3n − 13 pn , and therefore the Hilbert-Kunz series is
3 + (p3 − 4p)t
.
3(1 − pt)(1 − p3 t)
1.1.1
Some Product Formulas in Γ.
[HM93, Lemma 3.3] Suppose 1 ≤ i ≤ q and j ≥ 1. Then:
δi δqj = iδqj
(1.1)
δi δqj+1 = (i − 1)δqj + δqj+i
(1.2)
δi δqj−1 = (i − 1)δqj + δqj−i .
(1.3)
[HM93, Theorem 3.4] Suppose i < q. Then:
λi λqj = λqj+i .
If j 6= 0, λi λqj−1 = λqj−1−i .
10
(1.4)
(1.5)
(In particular, λ2q−1 = 1.)
[HM93, Theorems 3.7, 3.10]
If p|j, λq λqj = λq(j+1) .
(1.6)
If p|j + 1, λq λqj = λq(j−1) .
(1.7)
If p 6 |j and p 6 |j + 1, λq λqj = λq(j−1) + λq(j+1)−1 + λq(j+1) .
(1.8)
[HM93, Theorem 2.5] If 0 ≤ i ≤ j < p, then
min(i+j,2p−2−i−j)
λi λ j =
X
λk .
k=j−i
1.1.2
The Function φ.
Definition 1.5. φ is the linear operator on Γ whose values on the basis {λi }
are defined as follows:
φ(λ2i ) := λ2pi ,
φ(λ2i−1 ) := λ2pi−1 .
Proposition 1.6. φn (Γ) is a subring of Γ.
Proof: Immediate from [HM93, Corollary 3.16], since φn (Γ) is just the Eq
in [HM93, Definition 3.12] (with q = pn ).
Definition 1.7. Γn is the subgroup of Γ generated by the λi with 0 ≤ i < pn .
Note that Γn is actually a subring of Γ, by [HM93, Theorem 3.2].
11
Proposition 1.8. φn (Γ)Γn = Γ.
Proof: It suffices to show that λi ∈ φn (Γ)Γn , for all i. Divide i by q := pn ,
obtaining i = aq + b, with 0 ≤ b < q.
If a is even, then λi = λaq λb = φn (λa )λb ∈ φn (Γ)Γn .
If a is odd, then λi = λq(a+1)−1 λq−1−b = φn (λa )λq−1−b ∈ φn (Γ)Γn .
For later use we note:
Proposition 1.9. If u ∈ φn (Γ) and v ∈ Γn then α(uv) = α(u)α(v).
Proof: We may assume that u = φn (λi ) and v = λj , with j < pn . But
then the result is clear, as [HM93, Theorem 1.10] shows that both sides of
the identity are zero, unless i = j = 0, in which case both sides are equal to
1.
Our goal in the remainder of this section is to show that φ is a ring
homomorphism.
Lemma 1.10. λq−1 λqj = λq(j+1)−1 . In particular, λ2q−1 = λq−1 λq .
Proof: Apply (1.4), with i = q − 1.
Lemma 1.11. Suppose 1 ≤ j ≤ p − 2. Then
(1) λ2q−1 λq(j+1)−1 = λq(j−1) + λq(j+1)−1 + λq(j+1) ;
(2) λ2q−1 λqj = λqj−1 + λqj + λq(j+2)−1 .
12
Proof: λ2q−1 λq(j+1)−1 = (λq−1 λq )(λq−1 λqj ) = λq λqj , and (1.8) gives (1).
Also λ2q−1 λqj = λq−1 λq λqj = λq−1 (λq(j−1) + λq−1 λqj + λq(j+1) ), and lemma
1.10 gives (2).
In what follows, let µ0 , µ1 , . . . , µp−1 be λ0 , λ2q−1 , λ2q , λ4q−1 , λ4q , . . ..
Lemma 1.12.
(1) If 1 ≤ j ≤ p − 2, µ1 µj = µj−1 + µj + µj+1 .
(2) µ1 µp−1 = µp−2 .
Proof: If j is odd, µj = λq(j+1)−1 , while if j is even, µj = λqj . Lemma 1.11
then gives (1). When p = 2, (2) is just the identity (λ2q−1 )2 = λ0 . On the
other hand, when p 6= 2, λ2q−1 λq(p−1) = λq λq−1 λq(p−1) = λq λqp−1 = λq(p−1)−1 ;
this is precisely (2).
Theorem 1.13. If 0 ≤ i ≤ j < p, then
min(i+j,2p−2−i−j)
µi µj =
X
µk .
k=j−i
Proof: It suffices to prove the identity when i + j < p and to show that
µk µp−1 = µp−1−k , for k < p. Both of these results are proved inductively,
using lemma 1.12 as the base case. The calculations are precisely the same
as those of [HM93, Lemma 2.4, Theorem 2.5].
The above theorem and [HM93, Theorem 2.5] give us the following result:
Lemma 1.14. Suppose 0 ≤ i, j < p. Then φn (λi λj ) = φn (λi )φn (λj ).
13
Lemma 1.15. If u ∈ φn (Γ) and v ∈ Γn , then φ(uv) = φ(u)φ(v).
Proof: It suffices to show that φ(λ2qi λj ) = φ(λ2qi )φ(λj ) and φ(λ2qi−1 λj ) =
φ(λ2qi−1 )φ(λj ), for any i and any j < q := pn .
Consider the case when j is odd. Then:
φ(λ2qi )φ(λj ) = λ2pqi λpj+p−1
= λ2pqi+pj+p−1
= φ(λ2qi+j )
= φ(λ2qi λj ),
and similarly:
φ(λ2qi−1 )φ(λj ) = λ2pqi−1 λpj+p−1
= λ2pqi−pj−p
= φ(λ2qi−j−1 )
= φ(λ2qi−1 λj ).
The case when j is even is treated similarly.
Theorem 1.16. φ is a ring homomorphism.
Proof: We’ll show by induction on n that φ(uv) = φ(u)φ(v) for all u and
v in Γn . For n = 1 use lemma 1.14. Now suppose the assertion holds for
n ≥ 1, and let u, v ∈ Γn+1 . We may assume without loss of generality that
u = λa and v = λb , with 0 ≤ a, b < pn+1 . Then as in proposition 1.8 we may
write λa = φn (λi )λj with 0 ≤ i < p and 0 ≤ j < pn . Similarly, we write λb =
φn (λk )λl with 0 ≤ k < p and 0 ≤ l < pn . Then φ(uv) = φ φn (λi )φn (λk )λj λl .
14
Since both φn (Γ) and Γn are closed under multiplication, lemma 1.15 shows
that this is equal to φ φn (λi )φn (λk ) ·φ(λj λl ). Now, from lemma 1.14 and the
induction hypothesis it follows that φ(uv) = φ(φn (λi ))·φ(λj )·φ(φn (λk ))·φ(λl ),
and one more application of lemma 1.15 gives the desired result.
1.2
Regular Sequences
Definition 1.17. X is the set of all the sequences u = (u0 , u1 , u2 , . . .) with
entries un ∈ Γ with the following property: there is an m ∈ N such that
un ∈ Γn+m , for all n.
Note that X is a commutative ring under coordinatewise addition and
multiplication.
Definition 1.18. To each polynomial g ∈ k[x1 , . . . , xs ] we associate the element hgi = (hgi0 , hgi1 , hgi2 , . . .) ∈ X, with hgin as in definition 1.1.
Remark: We’ve seen in section 1.1 that
hgin =
q−1
X
(deg(xq1 , . . . , xqs , g i+1 ) − deg(xq1 , . . . , xqs , g i ))(−1)i λi ,
i=0
where q := pn . So hgin ∈ Γn , and hgi is in fact an element of X.
Example 1.19.
• hxi = (δ1 , δp , δp2 , . . .).
• hx + yi = hxihxi = (δ1 , pδp , p2 δp2 , . . .).
15
Definition 1.20. The shift operator is the endomorphism
S : X −→ X
(u0 , u1 , u2 , . . .) 7−→ (u1 , u2 , u3 , . . .)
Consider the embedding
Γ
,→
X
u 7−→ (u, φ(u), φ2 (u), φ3 (u), . . .).
Since φ is an endomorphism of Γ (theorem 1.16), this embedding is a ring
homomorphism, and X becomes a Γ-algebra. If we identify Γ with its image
in X, the restriction of the shift operator to Γ is φ.
Example 1.21. Shxi = δp hxi. In fact, a straightforward calculation shows
that φn (δp )δpn = δpn+1 . We illustrate with the case when p = 5. Let q = pn .
Then:
φn (δ5 )δq = φn (λ0 − λ1 + λ2 − λ3 + λ4 )(λ0 − λ1 + . . . + λq−1 )
= (λ0 − λ2q−1 + λ2q − λ4q−1 + λ4q )(λ0 − λ1 + . . . + λq−1 )
= (λ0 − λ1 + . . . + λq−1 )
−(λ2q−1 − λ2q−2 + . . . + λq )
+(λ2q − λ2q+1 + . . . + λ3q−1 )
−(λ4q−1 − λ4q−2 + . . . + λ3q )
+(λ4q − λ4q+1 + . . . + λ5q−1 )
= δ5q .
16
Definition 1.22. Let ZN be the ring of sequences of integers. J : X →
ZN is the function induced by α, namely the function (u0 , u1 , u2 , . . .) 7→
(α(u0 ), α(u1 ), α(u2 ), . . .).
Theorem 1.23. Let M be a finitely generated additive subgroup of X, with
S(M ) ⊆ Γ · M . Then J(Γ · M ) is a finitely generated subgroup of ZN , stable
under the shift operator S : ZN → ZN , (u0 , u1 , u2 , . . .) 7→ (u1 , u2 , u3 , . . .).
Proof: Since M is finitely generated, we can choose m so that un ∈ Γn+m ,
for all u in M , and all n. We may replace M by the larger finitely generated
additive group Γm · M , and therefore assume that M = Γm · M .
By proposition 1.8, Γ · M = φm (Γ)Γm · M = φm (Γ) · M . Now proposition
1.9 shows that α(φm+n (λi )un ) = 0 for all u ∈ M , unless i = 0. Hence
J(Γ · M ) = J(φm (Γ) · M ) = J(M ), and therefore is finitely generated.
To conclude the proof, note that J(Γ · M ) is stable under the shift operator: S(J(Γ · M )) = J(S(Γ · M )) ⊆ J(Γ · S(M )) ⊆ J(Γ · M ).
Corollary 1.24. Under the hypotheses of theorem 1.23, there is a linear
recursive relation α(un+l ) = c1 α(un+l−1 ) + · · · + cl α(un ), with ci ∈ Z, which
∞
X
α(un )tn is a
holds for all u ∈ M , for all n ≥ 0. In particular, the series
n=0
quotient of polynomials in t with coefficients in Z.
Proof: Since J(Γ · M ) is finitely generated and stable under S, there are
integers ci such that the restriction of S to J(Γ · M ) satisfies an identity
17
S l = c1 S l−1 + c2 S l−2 + · · · + cl I, where I is the identity map. The result
follows.
Definition 1.25. u ∈ X is regular if there is a finitely generated additive
subgroup M of X containing u with S(M ) ⊆ Γ · M .
Example 1.26. hxi is regular. In fact, we can take M to be the additive
subgroup generated by hxi, and example 1.21 shows that S(M ) ⊆ Γ · M .
For technical reasons we’ll have to allow real coefficients, so we’ll now
revisit definitions 1.17 and 1.25 in this new context.
Definition 1.27. Let ΓR := Γ ⊗Z R, and let XR be the ring of sequences
u = (u0 , u1 , u2 , . . .) with entries un ∈ ΓR with the property that there is an
m ∈ N such that un ∈ Γn+m ⊗Z R, for all n.
We’ll denote the shift operator in XR and the function α ⊗ R simply by
S and α, by abuse of notation.
Definition 1.28. u ∈ XR is R-regular if there is a finitely generated subspace M of XR containing u with S(M ) ⊆ Γ · M .
The same arguments used in theorem 1.23 and corollary 1.24 can be used
to prove the following proposition:
Proposition 1.29. If u ∈ XR is R-regular, then the series
∞
X
α(un )tn is a
n=0
quotient of polynomials with real coefficients.
18
Definition 1.30. A polynomial g is strongly rational (resp. strongly
R-rational) if hgi is regular (resp. R-regular).
From corollary 1.24 and proposition 1.29 we immediately obtain the following:
Theorem 1.31. If g is strongly rational or R-rational, then the Hilbert-Kunz
series of g is rational.
Remark: If the Hilbert-Kunz series is rational, then it can be written as
a quotient of polynomials with integer coefficients, since the Hilbert-Kunz
series itself has integer coefficients.
Note that if S(M ) ⊆ Γ·M and S(N ) ⊆ Γ·N , then S(M +N ) ⊆ Γ(M +N )
and S(M N ) ⊆ Γ · M N . Consequently, if u and v are regular, then so are
u + v and uv. Moreover, any u ∈ Γ is regular (just take M to be Γ). So
the set of regular sequences is a subalgebra of X. The same reasoning shows
that the set of R-regular sequences is a subalgebra of XR .
The remarks above also show the following:
Theorem 1.32. If g ∈ k[x1 , . . . , xs ] and h ∈ k[y1 , . . . , yt ] are strongly rational or R-rational, then so is g + h.
We’ll now outline the proof of our main result—the rationality of the
X
Hilbert-Kunz series of polynomials of the type g :=
Gi (xi , yi ) (and z d +
i
19
X
Gi (xi , yi )), where each Gi is homogeneous. The discussion above shows
i
that it suffices to prove that each Gi is strongly R-rational. For then g is
also strongly R-rational, and theorem 1.31 concludes the proof.
So let H ∈ k[x, y] be a homogeneous polynomial. In order to show that
H is strongly R-rational we need to analyze the action of the shift operator
on hHi.
We’ve seen in section 1.1 that
hHin =
q−1
X
(deg(xq , y q , H i+1 ) − deg(xq , y q , H i ))(−1)i λi ,
i=0
where q := pn . Now if we let ϕH ( qi ) :=
1
q2
deg(xq , y q , H i ) we can write
q−1 i i + 1
X
hHin =
q 2 ϕH
− ϕH
(−1)i λi .
q
q
i=0
(Remark: the
1
q2
in the definition of ϕH ( qi ) makes its value independent of
pi
the choice of q, i.e. ϕH ( qi ) = ϕH ( pq
).) Now
pq−1
hHin+1 =
X
2
(pq) ϕH
i=0
i + 1
pq
i − ϕH
(−1)i λi ,
pq
and this sum can be split up into p sums
kq+q−1
Sk :=
X
i=kq
i + 1
i (pq)2 ϕH
− ϕH
(−1)i λi
pq
pq
(0 ≤ k < p).
If k is even then
Sk =
q−1
X
(pq)2 (ϕH ( i+kq+1
) − ϕH ( i+kq
))(−1)i λi+kq
pq
pq
i=0
2
= p λkq
q−1
X
q 2 (ϕH ( i+kq+1
) − ϕH ( i+kq
))(−1)i λi .
pq
pq
i=0
20
If q 0 is any power of p and a ∈ [q 0 − 1], then let Tq0 |a ϕH (x) = ϕH
x+a
q0
. So
q−1 i + 1
i X
Sk = p λkq
q 2 Tp|k ϕH
− Tp|k ϕH
(−1)i λi .
q
q
i=0
2
Now, for any function ψ : [0, 1] → R, let hψi be the sequence whose nth entry
is
q−1 i X
i + 1
2
hψin :=
q ψ
−ψ
(−1)i λi .
q
q
i=0
Then hHi = hϕH i, and Sk = p2 φn (λk )hTp|k ϕH in .
Now suppose k is odd. Then
kq+q−1
Sk =
X
(pq)2 (ϕH ( i+1
) − ϕH ( pqi ))(−1)i λi
pq
i=kq
=
q−1
X
(pq)2 (ϕH ( kq+q−1−i
) − ϕH ( kq+q−i
))(−1)i λkq+q−1−i
pq
pq
i=0
2
= p λkq+q−1
q−1
X
q 2 (Tp|k ϕH (1 −
i+1
)
q
− Tp|k ϕH (1 − qi ))(−1)i λi .
i=0
Now if we let R : [0, 1] → [0, 1] be the reflection x 7→ 1 − x, we have
Sk = p2 φn (λk )hTp|k ϕH ◦ Rin . In conclusion,
ShHi = p2 λ0 hTp|0 ϕH i + λ1 hTp|1 ϕH ◦ Ri + λ2 hTp|2 ϕH i + · · · .
Similarly, if we apply the shift operator to hTq|a ϕH i, we obtain a linear
combination of hTp|0 (Tq|a ϕH )i, hTp|1 (Tq|a ϕH ) ◦ Ri, hTp|2 (Tq|a ϕH )i, etc., with
coefficients in Γ. The shift operator applied to Tq|a ϕH ◦ R, on the other
hand, yields a linear combination of hTp|0 (Tq|a ϕH ◦ R)i, hTp|1 (Tq|a ϕH ◦ R) ◦ Ri,
hTp|2 (Tq|a ϕH ◦ R)i, etc.. One can easily see that Tp|k (Tq|a ϕH ) = Tpq|k+ap ϕH
21
and Tp|k (Tq|a ϕH ◦ R) = Tpq|c ϕH ◦ R, where c := ap + p − k − 1. Thus, if we
take M to be the real vector space spanned by hHi and all the hTq|a ϕH i and
hTq|a ϕH ◦ Ri, then the discussion above shows that S(M ) ⊆ Γ · M . Our proof
then amounts to showing that M is finite dimensional. This motivates the
definition of p-fractals, in the next chapter, and our main result will follow
from the fact that ϕH is a p-fractal.
22
Chapter 2
p-Fractals and the Main
Theorem
2.1
p-Fractals
Let F r be the real algebra of continuous real valued functions defined on the
r dimensional cube [0, 1]r . For any q and any a = (a1 , . . . , ar ) ∈ [q − 1]r we
define an algebra homomorphism Tq|a : F r → F r as follows: if ϕ ∈ F r , then
Tq|a (ϕ)(x) := ϕ
x + a
q
, ∀x ∈ [0, 1]r .
Roughly speaking, what we’re doing is partitioning the cube [0, 1]r into q r
pieces, and each Tq|a (ϕ) is obtained by looking at the values of ϕ on a particular piece.
Note that T1|0 is the identity map, and Tq|a ◦ Tq0 |b = Tqq0 |a+bq , so the Tq|a
23
form a semigroup under composition, which will be denoted by T .
Definition 2.1. ϕ ∈ F r is a p-fractal if the subspace of F r spanned by all
the transforms of ϕ under the semigroup T is finite dimensional. Equivalently, ϕ is a p-fractal if it is contained in a finite dimensional subspace of F r
which is stable under the action of T . The set of all p-fractals is a subalgebra
Fpr of F r .
Example 2.2. The coordinate function Xi is a p-fractal, since the subspace
of F r spanned by 1 and Xi contains Xi and all its transforms. Hence all
polynomial functions are p-fractals.
Example 2.3 ([HM93]). There is a unique continuous map Dr : [0, 1]r →
k[x1 , . . . , xr ]
1 dim
, for all q and
[0, 1] such that Dr ( aq1 , . . . , aqr ) = r−1
a1
k
(x1 , . . . , xar r , Σxi )
q
all a ∈ [q]r . The subspace spanned by
• Dr
• (x1 , . . . , xr ) 7→ Dr (1 − x1 , x2 , . . . , xr )
•
Y
Xi , where S ⊂ {1, . . . , r} and |S| < r
i∈S
is stable under T , and therefore Dr is a p-fractal. For instance, suppose
p ≥ 3. Then p3 D4 x1q+1 , x2q+1 , x3q+1 , x4q+1 = 3D4 (x1 , x2 , x3 , x4 ) − s3 + s2 + 1,
where si is the ith symmetric polynomial in x1 , x2 , x3 , x4 ([HM93, cor 4.14]).
In section 1.2 we indicated that the key to proving the rationality of
X
X
Hilbert-Kunz series of polynomials of type
Gi (xi , yi ) and z d + Gi (xi , yi ),
i
24
i
with Gi homogeneous, is showing that any homogeneous polynomial H in
two variables is strongly rational. We also indicated that this could be accomplished by showing that the function ϕH , loosely “defined” by ϕH ( qi ) :=
1
q2
deg(xq , y q , H i ), is a p-fractal. To make this a little more precise, we’ll show
that there is a unique continuous function ϕH : [0, 1] → R whose value at
(i ∈ [q]) is
1
q2
i
q
deg(xq , y q , H i ), and that this function is a p-fractal. The rest of
this chapter will develop the machinery needed to prove that. We end this
section with some properties of p-fractals that will be used.
Definition 2.4. A reflection is a map R : [0, 1]r → [0, 1]r whose ith coordinate R(x)i is either xi or 1 − xi , for all i = 1, . . . , r.
Proposition 2.5. If ϕ : [0, 1]r → R is a p-fractal and R is a reflection, then
ϕ ◦ R is also a p-fractal.
Proof: By decomposing R we may assume that only one of the coordinates
is changed by R—so we may assume that R(x) = (1 − x1 , x2 , . . . , xr ). Then
Tq|a (ϕ ◦ R)(x) = ϕ ◦ R( x+a
)
q
= ϕ(1 −
x1 +a1 x2 +a2
r
, q , . . . , xr +a
)
q
q
= Tq|b (ϕ)(R(x))
where b = (q − 1 − a1 , a2 , . . . , ar ). Hence T (ϕ ◦ R) is contained in a finitedimensional subspace, namely the image of span(T ϕ) under the algebra morphism F r → F r , ψ 7→ ψ ◦ R.
25
Proposition 2.6. Let D : [0, 1] → [0, 1]r be the “diagonal map” x 7−→
(x, x, . . . , x), and suppose ϕ : [0, 1]r → R is a p-fractal. Then ϕ ◦ D is a
p-fractal.
Proof: Tq|a (ϕ ◦ D) = (Tq|D(a) ϕ) ◦ D for any q and any 0 ≤ a < q, and
therefore T (ϕ ◦ D) is contained in a finite dimensional subspace—the image
of span(T ϕ) under the morphism F r → F 1 , ψ 7→ ψ ◦ D.
The same argument shows a more general result:
Proposition 2.7. Let D : [0, 1]s → [0, 1]r be the map (x1 , . . . , xs ) 7−→
(xi1 , . . . , xir ), for some 1 ≤ ik ≤ s. If ϕ ∈ Fpr , then ϕ ◦ D ∈ Fps .
Let ϕ ∈ F r . Fix positive integers m1 , . . . , mr , and for each i = (i1 , . . . , ir )
with 0 ≤ ik < mk we define a function ϕ(i) ∈ F r as follows:
ϕ(i) (x) := ϕ
x + i
x r + ir 1
1
,...,
,
m1
mr
∀x ∈ [0, 1]r .
Proposition 2.8. If all the ϕ(i) are p-fractals, then ϕ is a p-fractal.
Proof: Let V be a finite dimensional subspace of F r which contains all
the ϕ(i) and is stable under the action of T . Let W consist of all the ψ ∈ F r
such that ψ (i) ∈ V , for all i. Then W is a finite dimensional subspace of F r
which contains ϕ. We’ll now show that W is stable under T .
Let ψ ∈ W and Tq|a ∈ T . We must show that (Tq|a ψ)(i) is an element of
26
V , for all i.
x + i
x r + ir 1
1
,...,
m1
mr
!
x1 +i1
xr +ir
+ a1
+ ar
m1
mr
,...,
q
q
(Tq|a ψ)(i) (x) = Tq|a ψ
= ψ
x + i + a m
xr + ir + ar mr 1
1
1 1
= ψ
,...,
qm1
qmr
We can write ik + ak mk = jk q + bk , with 0 ≤ jk < mk and 0 ≤ bk < q. So
!
x1 +b1
xr +br
+
j
+
j
r
1
q
q
(Tq|a ψ)(i) (x) = ψ
,...,
mr
m1
x + b
x r + br 1
1
(j)
= ψ
,...,
q
q
= Tq|b (ψ (j) )(x).
Hence (Tq|a ψ)(i) = Tq|b (ψ (j) ) and, since ψ ∈ W and V is stable under T ,
(Tq|a ψ)(i) ∈ V , giving the result.
2.2
Syzygy Gaps and δ
Let A be the polynomial ring k[x, y], and suppose H ∈ A is homogeneous.
As previously seen, the key to proving the strong R-rationality of H (and,
consequently, our main result) is to understand how deg(xq , y q , H k ) depends
on q and k. More generally one may ask how deg(xi , y j , H k ) depends on i, j
and k. We’ll see in this section how this number is related to the degrees of
the syzygies between xi , y j and H k . We’ll actually start with a more general
27
situation, looking at deg(F, G, H), where F, G, H ∈ A are nonzero homogeneous polynomials of degrees d1 , d2 , d3 , respectively. For now we also assume
that these polynomials have no common factor. So either (F, G, H) = A or
(F, G, H) is an (x, y)-primary ideal; thus A/(F, G, H) is a finite dimensional
vector space over k.
By the Hilbert Syzygy Theorem, the module of relations between F , G
and H, denoted by Syz(F, G, H), is free of rank 2. Let m ≤ n be the degrees
of its generators. Then there is an exact sequence
0 → A(−m) ⊕ A(−n) → A(−d1 ) ⊕ A(−d2 ) ⊕ A(−d3 ) → A → A/(F, G, H) → 0.
Let h(t) be the Hilbert series of A/(F, G, H). Then deg(F, G, H) is just
h(1), and this exact sequence will make it possible to express deg(F, G, H)
in terms of the degrees d1 , d2 , d3 , m and n. Using the fact that the Hilbert
series of A(−k) is tk /(1 − t)2 , from the exact sequence above we obtain:
h(t) =
1 − td1 − td2 − td3 + tm + tn
.
(1 − t)2
Differentiating (1 − t)2 h(t) and setting t = 1 yields m + n = d1 + d2 + d3 .
Differentiating one more time, setting t = 1, and dividing by 2, we get
28
h(1) =
1
(m(m − 1) + n(n − 1) − d1 (d1 − 1) − d2 (d2 − 1) − d3 (d3 − 1))
2
=
1 2
(m + n2 − d1 2 − d2 2 − d3 2 )
2
=
1
((m + n)2 + (m − n)2 − 2d1 2 − 2d2 2 − 2d3 2 )
4
=
1
(m − n)2
(2d1 d2 + 2d1 d3 + 2d2 d3 − d1 2 − d2 2 − d3 2 ) +
.
4
4
Definition 2.9. The syzygy gap of F , G, and H is the nonnegative integer
δ := n − m, where m ≤ n are the degrees of the generators of Syz(F, G, H).
We have proved the following:
Theorem 2.10. Let F, G, H ∈ k[x, y] be nonzero forms of degrees d1 , d2 , d3 ,
respectively, and suppose F , G and H have no common factor. Let m ≤ n
be the degrees of the generators of Syz(F, G, H), and let δ := n − m be the
syzygy gap. Then m + n = d1 + d2 + d3 and
deg(F, G, H) = Q(d1 , d2 , d3 ) +
δ2
,
4
1
where Q(d1 , d2 , d3 ) := (2d1 d2 + 2d1 d3 + 2d2 d3 − d1 2 − d2 2 − d3 2 ).
4
29
Note that theorem 2.10 shows that δ = d1 +d2 +d3 −2m, which motivates
the following definition:
Definition 2.11. Given any nonzero forms F , G and H in k[x, y], we define
δ(F, G, H) := deg F + deg G + deg H − 2m(F, G, H), where m(F, G, H) is the
smallest degree of a nontrivial relation between F , G and H.
Thus, if F , G and H have no common factor, δ(F, G, H) is the syzygy
gap of F , G and H.
In the rest of this section we’ll establish some basic properties of the
function δ. As before, let F , G and H be nonzero homogeneous polynomials
of degrees d1 , d2 , d3 , respectively.
Proposition 2.12. δ(P F, P G, P H) = deg P + δ(F, G, H), for any homogeneous polynomial P 6= 0.
Proof: Let d := deg P . Note that Syz(P F, P G, P H)(d) = Syz(F, G, H),
so that m(P F, P G, P H) = d + m(F, G, H). Therefore δ(P F, P G, P H) =
(d1 + d) + (d2 + d) + (d3 + d) − 2(m(F, G, H) + d) = d + δ(F, G, H).
Proposition 2.13. δ(F, G, H) ≥ 0, and δ(F, G, H) = 0 if and only if d1 +
d2 + d3 is even and there are no nontrivial relations between F, G, H of degree
1
(d
2 1
+ d2 + d3 ) − 1.
Proof: The first assertion is immediate if F , G and H have no common
factor, since δ(F, G, H) is the syzygy gap—nonnegative by definition. In case
30
F , G and H do have a common factor, we can write F = P F1 , G = P G1 , and
H = P H1 , where F1 , G1 and H1 have no common factor, and proposition
2.12 shows that δ(F, G, H) = deg P + δ(F1 , G1 , H1 ) > 0.
The second assertion is now clear: δ(F, G, H) equals 0 if and only if
m(F, G, H) ≥ 21 (d1 + d2 + d3 ), which is the same as saying that there are no
nontrivial syzygies of degree 12 (d1 + d2 + d3 ) − 1.
Proposition 2.14. Suppose d1 + d2 + d3 is even and assume that F , G, and
H have no common factor. Then
δ(F, G, H) = 2 dimk Syz(F, G, H) d1 +d2 +d3 −1 .
2
Proof: Using the same notation as in theorem 2.10, we can write
Syz(F, G, H) d1 +d2 +d3 −1 = (A(−m) ⊕ A(−n)) m+n −1 .
2
2
The dimension of the second summand in degree
dimk Syz(F, G, H) d1 +d2 +d3 −1 =
2
m+n
2
− 1 is 0, so
m+n
δ(F, G, H)
−m=
.
2
2
Proposition 2.15. deg(F p , Gp , H p ) = p2 deg(F, G, H).
Proof: We may assume without loss of generality that k = kp . k[x, y] is free
of rank p2 over k[xp , y p ], so deg(F p , Gp , H p ) = p2 dimk k[xp , y p ]/(F p , Gp , H p ).
But one can easily see that k[xp , y p ]/(F p , Gp , H p ) and k[x, y]/(F, G, H) have
31
the same dimension.
of k[x, y]/(F, G, H).
For suppose u1 , . . . , us ∈ k[x, y] represent a basis
P p
P 1/p
If
ci ui ∈ (F p , Gp , H p )k[xp , y p ], then
ci ui ∈
(F, G, H), and all the ci must be zero. So up1 , . . . , ups represent linearly independent elements of k[xp , y p ]/(F p , Gp , H p ). A similar argument shows that
they span this space.
Proposition 2.16. δ(F p , Gp , H p ) = pδ(F, G, H).
Proof:
In view of proposition 2.12 we may assume that F , G and H
have no common factor. Now theorem 2.10 and proposition 2.15 show that
δ(F p , Gp , H p )2 = 4 deg(F p , Gp , H p ) − 4Q(pd1 , pd2 , pd3 ) = 4p2 deg(F, G, H) −
4p2 Q(d1 , d2 , d3 ) = p2 δ(F, G, H)2 , giving the desired result.
Proposition 2.17. Let P ∈ k[x, y] be a nonzero homogeneous polynomial,
and suppose P and H have no common factor. Then δ(P F, P G, H) =
δ(F, G, H).
Proof: If (α, β, γ) ∈ Syz(F, G, H)d , then (α, β, P γ) is a syzygy between
P F , P G, and H, of degree d + deg P . Thus m(P F, P G, H) ≤ m(F, G, H) +
deg P . On the other hand, if (α, β, γ) ∈ Syz(P F, P G, H)d , then P divides
γ
γ, and (α, β, P ) is a relation between F , G and H, of degree d − deg P .
Therefore m(P F, P G, H) = m(F, G, H) + deg P , and the result follows.
32
Proposition 2.18. Suppose d1 ≥ d2 + d3 , and suppose G and H have no
common factor. Then δ(F, G, H) = d1 − d2 − d3 . Similar results hold when
d2 ≥ d1 + d3 or d3 ≥ d1 + d2 .
Proof: It’s easy to see that under this hypothesis (0, −H, G) is a syzygy
between F , G and H of minimal degree. Therefore m(F, G, H) = d2 + d3 ,
and δ(F, G, H) = d1 − d2 − d3 .
We’ll now investigate how δ(F, G, H) changes as we multiply one of its
entries, say H, by a linear form `. As a first rather obvious remark, note that
δ(F, G, `H) is different from δ(F, G, H), since δ(F, G, H) ≡ deg F + deg G +
deg H (mod 2). In fact, we have the following:
Proposition 2.19. Let P ∈ k[x, y] be a nonzero homogeneous polynomial.
Then
|δ(F, G, P H) − δ(F, G, H)| ≤ deg P.
In particular, if ` is a linear form then δ(F, G, `H) = δ(F, G, H) ± 1.
This result follows directly from the next proposition and definition 2.11.
Proposition 2.20. Let P be a nonzero form. Then
m(F, G, H) ≤ m(F, G, P H) ≤ m(F, G, H) + deg P.
Proof:
Let m := m(F, G, H) and m+ := m(F, G, P H). Every syzygy
(α, β, γ) ∈ Syz(F, G, H)d gives (P α, P β, γ) ∈ Syz(F, G, P H)d+deg P , which
33
shows that m + deg P ≥ m+ . Conversely, any (α, β, γ) ∈ Syz(F, G, P H)d
gives (α, β, P γ) ∈ Syz(F, G, H)d , implying that m+ ≥ m.
It is evident that similar results relate m(F, G, H) to m(P F, G, H) and
m(F, P G, H); consequently similar results relate δ(F, G, H) to δ(P F, G, H)
and δ(F, P G, H).
Now let P ∈ k[x, y] be a nonzero homogeneous polynomial of degree
d, and assume that F , G, and P H have no common factor. Let δ− :=
δ(F, G, H), δ := δ(F, G, P H) and δ+ := δ(F, G, P 2 H). Multiplication by P
gives us a surjective map:
(F, G, H)
(F, G, P H)
P
−→
(F, G, P H)
(F, G, P 2 H)
Therefore deg(F, G, P H) − deg(F, G, H) ≥ deg(F, G, P 2 H) − deg(F, G, P H),
and using 2.10 and simplifying we obtain
2
2
δ−
+ δ+
≤ 2(δ 2 + d2 ).
Consider now the case when P = `, a linear form, and suppose we have
a situation where both δ− and δ+ are greater than δ. Then δ− and δ+ are
≥ δ +1, and the inequality derived above implies that δ = 0. We have proved
the following:
Proposition 2.21.
• Let P ∈ k[x, y] be a nonzero form of degree d, and suppose F , G and
P H have no common factor. Then
δ(F, G, H)2 + δ(F, G, P 2 H)2 ≤ 2(δ(F, G, P H)2 + d2 ).
34
• Let ` be a linear form, and suppose F , G and `H have no common factor. If δ(F, G, H) and δ(F, G, `2 H) are both greater than δ(F, G, `H),
then δ(F, G, `H) = 0.
Once again it’s evident that similar results are obtained using F , P G and
H or P F , G and H.
Lemma 2.22. Suppose F , G and H` have no common factor, and (α, β, γ)
is a syzygy between F , G and H of minimal degree. Then m(F, G, H) =
m(F, G, H`) if and only if δ(F, G, H) = 0 or ` divides γ.
Proof:
Suppose there is a nontrivial syzygy (α0 , β 0 , γ 0 ) of degree m :=
m(F, G, H) between F , G and H`. Then (α0 , β 0 , γ 0 `) is a degree m relation
between F, G, H, which means it has to be a constant multiple of (α, β, γ),
unless the other generator of Syz(F, G, H) also has degree m. Hence either `
divides γ or δ(F, G, H) = 0.
γ
Conversely, suppose that ` divides γ. Then (α, β, ) is a relation between
`
F , G and H` of degree m, and proposition 2.20 shows that m(F, G, H`) has
to be m. On the other hand, suppose ` does not divide γ, but δ(F, G, H) = 0.
If (α0 , β 0 , γ 0 ) is the other generator of Syz(F, G, H), we can find a constant c
such that γ 0 + cγ is a multiple of `. Therefore (α0 + cα, β 0 + cβ,
γ 0 + cγ
) is
`
a degree m syzygy between F , G and H`, and, as before, m(F, G, H`) = m.
35
Proposition 2.23. Let F, G, H ∈ k[x, y] be nonzero homogeneous polynomials, and let `1 and `2 be relatively prime linear forms, such that F , G and
H`1 `2 have no common factor. Suppose that δ(F, G, H) = δ(F, G, H`1 `2 ) and
δ(F, G, H`1 ) = δ(F, G, H`2 ). Then either δ(F, G, H) = 0 or δ(F, G, H`1 ) =
0.
Proof: Suppose δ(F, G, H) > 0, and let (α, β, γ) be a syzygy between F ,
G and H of minimal degree m. The hypothesis immediately implies that
m(F, G, H`1 `2 ) = m + 1, and m(F, G, H`1 ) = m(F, G, H`2 ) = m0 , which is
either m or m+1, by proposition 2.20. If m0 = m then lemma 2.22 shows that
`1 `2 divides γ, which contradicts the hypothesis, since (α, β, γ/`1 `2 ) would be
a relation between F , G, and H`1 `2 of degree m. So m0 = m + 1, and neither
`1 nor `2 divides γ. Now (`1 α, `1 β, γ) is a relation between F , G and H`1 of
minimal degree, m(F, G, H`1 `2 ) = m(F, G, H`1 ), and `2 does not divide γ,
so another application of lemma 2.22 shows that δ(F, G, H`1 ) = 0.
Similar reasoning gives the following result:
Proposition 2.24. Let F, G, H ∈ k[x, y] be nonzero homogeneous polynomials, and let `1 and `2 be relatively prime linear forms, such that F `1 , G`2 and
H have no common factor. Suppose that δ(F, G, H) = δ(F `1 , G`2 , H) and
δ(F `1 , G, H) = δ(F, G`2 , H). Then either δ(F, G, H) = 0 or δ(F `1 , G, H) =
0.
36
2.3
The Functions δH and ϕH
Let H ∈ k[x, y] be a nonzero homogeneous polynomial. For any q and any
a ∈ [q] we define:
δH
a
q
:=
1
· δ(xq , y q , H a )
q
and
ϕH
a
q
:=
1
· deg(xq , y q , H a ).
q2
Note that proposition 2.16 guarantees that these values are independent of
the choice of q. Also, proposition 2.19 shows that |δH ( a+i
) − δH ( aq )| ≤ deg H ·
q
i
,
q
for all a ∈ [q − i]. So if we let [0, 1]p be the set of rational numbers in [0, 1]
which can be written as aq , we have the following result:
Proposition 2.25 (Lipschitz Property). If x, y ∈ [0, 1]p , then
|δH (x) − δH (y)| ≤ deg H · |x − y|.
So the function δH is uniformly continuous, and can be uniquely extended
by continuity to the whole interval [0, 1]:
Definition 2.26. Let H ∈ k[x, y] be a nonzero homogeneous polynomial.
Then δH : [0, 1] → [0, ∞) is the continuous function whose value at
is δH ( aq ) :=
1
q
· δ(xq , y q , H a ).
37
a
q
∈ [0, 1]p
2
Now, theorem 2.10 shows that ϕH differs from δH
by a polynomial, and
therefore it can also be extended to a continuous function defined on [0, 1]:
Definition 2.27. ϕH : [0, 1] → [0, ∞) is the continuous function whose value
at
a
q
∈ [0, 1]p is ϕH ( aq ) :=
1
q2
· deg(xq , y q , H a ).
We conclude this section looking at some examples of functions δH .
Example 2.28. Let p = 3 and let be algebraic over Z/3Z, with 2 +2+2 =
0. The graph of the function δH , where H = xy(x + y)(x + y), is shown
i
below. It was obtained by computing δH ( 729
) (0 ≤ i ≤ 729) with the software
Macaulay 2, and plotted on Maple.
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
δH (x) (0 ≤ x ≤ 1)
It’s clear from the graph that δH (x) = 4x − 2 on the interval [ 21 , 1]. This
is expected: if
δH ( aq ) =
1
q
a
q
≥
1
2
then deg(H a ) ≥ 2q, and proposition 2.18 shows that
· δ(xq , y q , H a ) = 1q (4a − 2q) = 4( aq ) − 2. So T3|2 δH (x) = 34 x + 23 .
Examining the blow-up of the region [ 31 , 23 ], it appears that T3|1 δH = 13 δH :
38
0.6
0.5
0.4
0.3
0.2
0.1
0
0.35
0.4
0.45
0.5
0.55
0.6
0.65
δH (x) ( 1
≤x≤ 2
)
3
3
Finally, a look at the region [0, 13 ] indicates that T9|0 δH = 91 δH , T9|1 δH =
1
|4X
9
− 2|, and T9|2 δH = 19 δH ◦ R, where R is the reflection map t 7→ 1 − t:
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
δH (x) (0 ≤ x ≤ 1
)
3
So the subspace of F 1 spanned by δH , δH ◦ R, |4X − 2|, 1 and X appears
to be stable under the action of the semigroup T ; in fact this is true.
39
Example 2.29. The following is the graph of δH , where H = x4 + xy 3 and
p = 5.
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
δH (x) (0 ≤ x ≤ 1)
The blow-up of the region [ 15 , 25 ] indicates that T5|1 δH = 15 δH :
0.4
0.3
0.2
0.1
0
0.25
0.3
0.35
0.4
δH (x) ( 1
≤x≤ 2
)
5
5
The most interesting part, however, is the interval [0, 15 ]. A blow-up of this
40
region indicates that T25|0 δH = T25|3 δH =
1
δ ,
25 H
T25|1 δH = T25|4 δH =
1
δ ◦ R,
25 H
and T25|2 δH = 51 T5|0 δH . Again, this can all be proven rigorously.
0.08
0.06
0.04
0.02
0
0.05
0.1
0.15
0.2
δH (x) (0 ≤ x ≤ 1
)
5
With the above examples we intend to give the reader an idea of what
the functions δH look like, and a sense of why it’s reasonable to expect them
(and consequently the ϕH ) to be p-fractals. However, the proof of that fact
will require the study of higher dimensional fractals, which will be introduced
in the next section.
2.4
The Function δa
Fix r, and let `1 , . . . , `r be pairwise prime linear forms. In order to simplify
notation, we write:
` := `1 `2 · · · `r .
41
If a = (a1 , . . . , ar ) is a nonnegative integer vector, then
`a := `a11 `a22 · · · `ar r .
As in the previous section, we denote by [0, 1]p the subset of [0, 1] which
consists of rational numbers that can be written as aq , where q is a power of
p. For any other set S of real numbers we define Sp likewise.
Let a = (F , G) be a zero-dimensional ideal in A := k[x, y]/(`), where F
and G are the classes of homogeneous polynomials F and G in A. Moreover,
suppose a is non-principal. For any aq ∈ [0, 1]rp we define
δa
a
q
:=
1
· δ(F q , Gq , `a ).
q
Note that proposition 2.16 assures that this value doesn’t depend on the
choice of q. Note also that δa only depends on a, and not on the particular
choice of the generators F and G, or the choice of class representatives.
Remark: Note that this wouldn’t be true if a was principal. For, in this
case, suppose F is the generator, and let m := deg G−deg F . We can assume
that G = 0, by modifying G, if necessary, by a multiple of F . Then `|G and
`a |Gq . It follows that the generators of Syz(F q , Gq , `a ) have degrees q deg G
P
P
and q deg F + ai , and that 1q · δ(F q , Gq , `a ) = |m − aqi |, which is not quite
independent of the degree of G.
Example 2.30. Let p = 3 and let be algebraic over Z/3Z, with 2 + 2 +
2 = 0. The following diagram shows the values of the function (i, j) 7→
42
δ(x81 , y 81 , xi y i (x + y)j (x + y)j ), for 0 ≤ i, j ≤ 81 and i + j ≤ 81. The zeros
of the function were replaced by dots, to aid visualizing the patterns and
symmetries. On the region not shown (i.e. i + j > 81) the behavior of the
function is simple: it’s simply the map (i, j) 7→ 2(i + j − 81).
.
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40
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60
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80
If we set a = (x, y), `1 = x, `2 = y, `3 = x + y and `4 = x + y and define
δa as above, then what is shown is simply the values of 81 · δa (x, y, z, w) on
43
1
the lattice ( 81
Z ∩ [0, 1])4 , restricted to the plane defined by x = y and z = w.
Notice how we can “see” the graph of example 2.28 by looking at the
“SW-to-NE” diagonal of this diagram.
If we write εi := (0, . . . , 0, 1, 0, . . . , 0), where the 1 appears in the ith spot,
i
then proposition 2.19 shows that δa ( a+ε
) − δa ( aq ) = ± 1q , for all a ∈ [q]r with
q
ai < q. Repeated applications of this result gives us the following:
Proposition 2.31 (Lipschitz Property). If x, y ∈ [0, 1]rp , then
|δa (x) − δa (y)| ≤ d(x, y),
where d(x, y) is the taxi cab distance between x and y, given by
r
X
d(x, y) :=
|xi − yi |.
i=1
The function δa is therefore uniformly continuous, and can be extended by
continuity to the whole r-dimensional cube [0, 1]r :
Definition 2.32. δa : [0, 1]r → [0, ∞) is the unique continuous function with
δa
a
q
:=
1
· δ(F q , Gq , `a ),
q
for all q and all a ∈ [q]r .
Corollary 2.33 (Linearity). Let x ∈ [0, 1]r and y = x + tεi , with t + xi ∈
[0, 1], and suppose δa (y) − δa (x) = d(x, y) = |t|. Then δa is linear on the line
segment xy. Explicitly, δa (z) = δa (x) + d(z, x), for all z ∈ xy.
44
The results derived in section 2.2 can be used to prove certain “convexity”
properties of the function δa , which will, loosely speaking, show that δa is
completely determined once we know where it vanishes.
Lemma 2.34. Let a ∈ [q]r , and suppose 0 < ai < q and δa ( aq ) > 0. Then
i
i
either δa ( a+ε
) < δa ( aq ) or δa ( a−ε
) < δa ( aq ).
q
q
Proof: Proposition 2.21.
In terms of the function δa , proposition 2.23 can be rephrased as follows:
Lemma 2.35. Suppose δa ( aq ) = δa (
a+εi +εj
)
q
i
and δa ( a+ε
) = δa (
q
a+εj
)
q
for some
a ∈ [q]r and 0 ≤ i < j ≤ r, with ai , aj < q. Then either δa ( aq ) = 0 or
i
δa ( a+ε
) = 0.
q
In what follows, C is the r-dimensional cube [0, 1]r , and for each q, Cq is
the set of points of C of the form aq , with a ∈ [q]r . Thus C1 ⊂ Cp ⊂ Cp2 ⊂ . . .,
and C1 is just {0, 1}r , the set of corners of C.
Lemma 2.36. Suppose δa |C1 attains a “local minimum” at the corner u, in
the sense that the values of δa at all corners adjacent to u are greater than
δa (u). Moreover, suppose δa (u) > 0. Then
(∗)
δa (x) = δa (u) + d(x, u),
for all x ∈ C. In particular δa > 0 everywhere.
45
Proof: From proposition 2.19 we know that δa (v) = δa (u)+1 for any corner
v adjacent to u. Hence corollary 2.33 shows that (∗) holds for all the points
of the edges containing u.
Now we’ll show that (∗) is true for all points of Cq . Since q is arbitrary and
the function x 7→ δa (u) + d(x, u) is continuous, that will allow us to conclude
that the assertion holds for all x ∈ C. Suppose that (∗) fails for one or more
points of Cq . Among all such points, choose a point x whose distance to u is
minimum. From the first paragraph, we know that x does not lie in any of the
edges containing u, so at least two coordinates of x and u must be different.
Say ui 6= xi and uj 6= xj . Let y := x ± εqi , whichever is closer to u, and define
ε
ε
z := x ± qj similarly. Moreover, define w := x ± εqi ± qj , it being understood
that we make the same choice of signs we made when we defined y and z.
Since y, z and w are closer to u than x is, we have δa (y) = δa (u)+d(y, u) := s,
δa (z) = δa (u) + d(z, u) = s and δa (w) = δa (u) + d(w, u) = s − 1q . Lemma 2.35
now shows that δa (x) must equal s + 1q , and (*) holds for x, contradicting
our assumption.
Theorem 2.37. Notation as above. Let Z := {z ∈ C | δa (z) = 0}.
• If Z is empty, then we are in the situation of lemma 2.36.
• If Z is nonempty then δa (x) is the taxi-cab distance of x from the set
Z, for all x ∈ C.
46
Proof: If Z is empty, then there must be a corner u satisfying the hypothesis of the previous lemma, and there’s nothing to show. So suppose Z is
nonempty. As in the proof of lemma 2.36, by continuity it suffices to show
the result for x ∈ Cq . We argue by induction on ψ(x) := qδa (x). If ψ(x) = 0,
then x is in Z, and the result is trivial. If ψ(x) > 0 then there is a y ∈ Cq
adjacent to x with ψ(y) < ψ(x), so the theorem follows by induction. In
fact, suppose that δa (y) > δa (x), for all y adjacent to x. Then lemma 2.34
shows that x must be a corner and, since δa (x) > 0, x satisfies the hypothesis of lemma 2.36. So δa would be > 0 on the whole cube C, which is a
contradiction.
The same argument used to prove lemma 2.36 can be used to show the
following:
Theorem 2.38. Suppose δa |C1 attains a “local maximum” at the corner u.
Then δa (x) = δa (u) − d(x, u), for all x ∈ C with d(x, u) ≤ δa (u).
In the remainder of this section we’ll show that the function δa is a pfractal, for any homogeneous two generator, non-principal, zero-dimensional
ideal a in A := k[x, y]/(`). Let A be the set of all such ideals. We’ll start by
defining two notions of equivalence on A.
Definition 2.39. Two ideals a, b ∈ A are δ-equivalent if δa = δb .
Proposition 2.17 shows that δa does not change if we replace F and G by
47
U F and U G, where U is a homogeneous polynomial prime to each `i . This
suggests the following definition:
Definition 2.40. Let a, b ∈ A. We say that a is equivalent to b (Notation:
a ∼ b) if there exist U and V homogeneous, prime to each `i , such that
U a = V b.
The following proposition is an immediate consequence of proposition
2.17:
Proposition 2.41. If a, b ∈ A and a ∼ b, then a and b are δ-equivalent. Lemma 2.42. Let a = (F , G) ∈ A. Then a is equivalent to some b =
(F ∗ , G∗ ) ∈ A, with deg F ∗ ≤ r.
In the proof of this lemma we’ll need the following elementary result:
Lemma 2.43. If F, G ∈ k[x, y] are homogeneous polynomials with no common factor, then (x, y)deg F +deg G−1 ⊆ (F, G).
Proof: Let d1 := deg F , d2 := deg G, and let Vi be the space of homogeneous polynomials in x and y of degree i. Consider the linear map
Vd2 −1 × Vd1 −1 → Vd1 +d2 −1
(A, B) 7→ AF + BG
Since F and G have no common factor, this map is one-to-one. So it’s also
onto, and the lemma follows.
48
Proof of Lemma 2.42: The lemma will be proved by induction on m :=
min{deg F, deg G}. Choose H ∈ k[x, y] of degree ≤ 2 not divisible by any
of the `i . When the field k is infinite, one can obviously find such an H of
degree 1. If k is finite, let x2 + ax + b ∈ k[x] be an irreducible quadratic
polynomial, and set H = x2 + axy + by 2 .
If m ≤ r there’s nothing to prove. So suppose m > r. Then deg F > r
and deg G > r, and both F and G are in the ideal (H, `), by lemma 2.43. So
we may write F = F1 H and G = G1 H, for some F1 , G1 ∈ k[x, y]. Thus
a = H(F1 , G1 ) ∼ (F1 , G1 ),
and the lemma follows by induction.
Lemma 2.44. Let a = (F , G) as above, and suppose m := deg F −deg G ≥ r.
Then δa is linear. Explicitly, if B := {i | `i divides G}, then
δa = m +
X
Xi −
i∈B
X
Xi ,
i6∈B
where the Xi are the coordinate functions.
Proof: Let q be a power of p, and a ∈ [q]r . Let B := {i | `i divides G}, `aB :=
Y
`ai i and `aC := `a /`aB . Then `aB divides Gq , `aB and F q have no common
i∈B
factor, and `aC and Gq /`aB have no common factor. Therefore proposition
2.17 shows that
δa
a
q
=
1
1
· δ(F q , Gq , `a ) = · δ(F q , Gq /`aB , `aC ),
q
q
49
and since deg F q ≥ deg Gq /`aB + deg `aC , proposition 2.18 gives us:
δa ( aq ) =
1
(deg F q
q
− deg Gq /`aB − deg `aC )
= deg F − deg G +
Xa
Xa
i −
i
q
q.
i∈B
This holds for all q and all a ∈ [q]r , so δa = m +
X
i∈B
i6∈B
Xi −
X
Xi , by continuity.
i6∈B
In view of lemmas 2.42 and 2.44, we have:
Theorem 2.45. If k is finite, then there are only finitely many nonlinear
functions δa .
Corollary 2.46. If k is finite, then the δa span a finite dimensional subspace
of F r .
Definition 2.47. Let C := (F, G, H) be a triple of nonzero homogeneous
polynomials in k[x, y], with dim(F, G, H`) = 0. Given aq ∈ [0, 1]rp we define:
a
1
δC
:= · δ(F q , Gq , H q `a ).
q
q
This definition doesn’t depend on the choice of q, and yields a Lipschitz
function, which can therefore be uniquely extended to a continuous function
δC : [0, 1]r → [0, ∞).
Lemma 2.48. The colon ideal ((F, G) : H) can be generated by 2 homogeneous polynomials, U and V . Let b be the ideal (U , V ) of A. Then δC = δb
P
if b is non-principal; otherwise δC = |m − Xi |, for some integer m.
50
Proof:
The first assertion follows from the fact that Syz(F, G, H) has
two homogeneous generators. To prove the second, we must consider two
cases. First, suppose the colon ideal is non-principal. Then U H and V H
have degrees equal to the degrees of the generators of Syz(F, G, H), and
so | deg U − deg V | = δ(F, G, H), while deg U + deg V = deg F + deg G −
deg H. In view of the remark made at the beginning of this section, it
now suffices to show that δ(F q , Gq , H q `a ) = δ(U q , V q , `a ), for all q and all
a ∈ [q]r . Now ((F q , Gq ) : H q ) = (U q , V q ), and so ((F q , Gq , H q `a ) : H q ) =
(U q , V q , `a ). Then multiplication by H q maps k[x, y]/(U q , V q , `a ) bijectively
onto (F q , Gq , H q )/(F q , Gq , H q `a ). It follows that
deg(U q , V q , `a ) = deg(F q , Gq , H q `a ) − q 2 deg(F, G, H).
The result now follows from theorem 2.10.
Suppose, on the other hand, that the colon ideal is principal. Then the
greatest common divisor, U , of F and G generates this ideal. So ((F q , Gq ) :
H q `a ) is also principal, with generator U q . Then Syz(F q , Gq , H q `a ) has generators of degrees deg(U q H q `a ) and deg(F q Gq /U q ). It follows easily that
P
δC = |m − Xi |, where m := deg F + deg G − 2 deg U − deg H.
Theorem 2.49. The subspace V of F r generated by the functions δa and
P
functions of the type |m −
Xi |, with m ∈ Z, is stable under the action
of the semigroup T . Furthermore, if k is finite then each δa and δC is a
p-fractal.
51
Proof: Let Tq0 |b ∈ T , and ψ := Tq0 |b (δa ), where a = (F , G). Then for any
q and any a ∈ [q]r ,
ψ( aq ) = δa ( a+bq
)
qq 0
0
=
1
qq 0
=
1
q0
0
0
· δ(F qq , Gqq , `a+bq )
· δC ( aq ),
0
where C = (F q , Gq , `b ). By continuity, ψ =
1
q0
· δC , and lemma 2.48 shows
P
that ψ ∈ V . Moreover, the transforms of functions of the type |m− Xi | are
scalar multiples of functions of the same kind. This proves the first assertion.
P
Note that |m − Xi | is linear on [0, 1]r for any m ≤ 0 or m ≥ r. So all
those functions are contained in the subspace spanned by 1, the coordinate
P
functions Xi , and |m − Xi |, with 0 < m < r. Thus, in view of corollary
2.46, if k is finite then the subspace V is finite dimensional and stable under
the action of T , and hence each δa and δC is a p-fractal.
Definition 2.50. Let a = (F , G) ∈ A. For any q and any a ∈ [q]r , define
ϕa
a
q
:=
1
· deg(F q , Gq , `a ).
2
q
Theorem 2.10 shows that this function differs from δa 2 by a polynomial.
Hence it can be uniquely extended to a continuous function ϕa : [0, 1]r →
[0, ∞).
Corollary 2.51. If k is finite, then ϕa is a p-fractal.
52
2.5
The Main Theorem
We can now use the results of the previous section to show that the functions
δH and ϕH are p-fractals.
Theorem 2.52. Suppose k is finite, and let H ∈ k[x, y] be a nonzero homogeneous polynomial. Then δH is a p-fractal.
Proof: By replacing k by a larger field, if necessary, we can assume that
mr
1
H factors as a product of linear forms in k[x, y]. Explicitly, H = `m
1 · · · `r .
Q
Let a := (x, y) ⊆ k[x, y]/( `i ), and define δa as in section 2.4. Also, let ψ
be the continuous function [0, 1]r → [0, ∞) whose value at a/q ∈ [0, 1]rp is
1
q
1 a1
r ar
· δ(xq , y q , `m
· · · `m
).
1
r
Using the notation introduced before proposition 2.8, we can easily see
that ψ (0) = δa , while for any i 6= (0, . . . , 0), ψ (i) is a function of the type δC .
Q
(More precisely, ψ (i) = δC , for C = (x, y, `ikk ).) Thus ψ is a p-fractal, by
proposition 2.8. Now δH = ψ ◦ D, where D : [0, 1] → [0, 1]r is the “diagonal
map” x 7→ (x, x, . . . , x). The result then follows from proposition 2.6.
Theorem 2.53. If k is finite and H ∈ k[x, y] is a nonzero homogeneous
polynomial, then ϕH is a p-fractal.
Proof: Immediate from theorem 2.52, in view of the fact that ϕH differs
2
from δH
by a polynomial, by theorem 2.10.
53
We now have the necessary ingredients to complete the proof of our main
result.
Definition 2.54. If ϕ : [0, 1] → R is a continuous function, then hϕi ∈ XR
is the sequence whose nth entry is
q−1 i X
i + 1
2
q ϕ
hϕin :=
−ϕ
(−1)i λi ,
q
q
i=0
where q := pn .
The calculation made in the end of section 1.2 shows the following:
Proposition 2.55. Let R : [0, 1] → [0, 1] be the reflection map x 7→ 1 − x.
Then
Shϕi = p2 (λ0 hTp|0 ϕi + λ1 hTp|1 ϕ ◦ Ri + λ2 hTp|2 ϕi + · · · + λp−1 hTp|p−1 ϕi).
Corollary 2.56. Let M be the subspace of XR spanned by all the hTq|a ϕi
and hTq|a ϕ ◦ Ri, with Tq|a ∈ T . Then S(M ) ⊆ Γ · M . In particular, if ϕ is a
p-fractal, then hϕi is R-regular.
Proof: The corollary follows immediately from the previous proposition,
in view of the facts that the set of transformations Tq|a is closed under composition, and that for any Tq|a , Tq0 |b ∈ T , Tq|a (Tq0 |b f ◦ R) = Tqq0 |c f ◦ R, where
c := bq + q − a − 1.
Now suppose k is finite and H ∈ k[x, y] is a nonzero homogeneous polynomial. By theorem 2.53, ϕH is a p-fractal. The previous corollary then
54
shows that hHi = hϕH i is R-regular, and H is strongly R-rational. We have
proved the following theorem:
Theorem 2.57. Let k be a finite field, and H ∈ k[x, y] a nonzero homogeneous polynomial. Then H is strongly R-rational.
In view of theorem 1.32, the previous theorem gives us the result we were
looking for:
Theorem 2.58. Let k be a finite field, and let Gi ∈ k[xi , yi ] be nonzero hoX
X
mogeneous polynomials. Then
Gi and z d +
Gi are strongly R-rational,
i
i
and in particular their Hilbert-Kunz series are rational.
2.6
Other p-Fractals
Note: throughout this section we’ll assume that the field k is finite.
The functions ϕH , unlike the δH , can be easily generalized for arbitrary
polynomials in more than two variables. Let g ∈ k[x1 , . . . , xs ] be a nonzero
polynomial (not necessarily homogeneous). To simplify notation, write x :=
x1 , . . . , xs and xq := xq1 , . . . , xqs . For a/q ∈ [0, 1]p , we set ϕg ( aq ) = 1s ·
q
deg(xq , g a ). This is independent of the choice of q. The exact sequence
g
k[x]/(xq , g a ) −→ k[x]/(xq , g a+1 ) −→ k[x]/(xq , g) −→ 0
55
shows that 0 ≤ ϕg ( a+1
) − ϕg ( aq ) ≤ ϕg ( 1q ). If q = pn , then ϕg ( 1q ) = en (g)/q s ,
q
and it’s easy to see that ϕg ( 1q ) ≤ constant/q. So ϕg is a Lipschitz function,
and it extends to a continuous function [0, 1] → [0, ∞):
Definition 2.59. ϕg : [0, 1] → R is the continuous function whose value at
a
q
∈ [0, 1]p is 1s · deg(xq , g a ).
q
If ϕg is a p-fractal, then the same arguments we have used (with some
obvious modifications) show that g is strongly R-rational, and therefore the
Hilbert-Kunz series of g is rational.
We have proved that ϕg is a p-fractal when g is a form in two variables,
and as a matter of fact, Monsky has recently showed that the same is true
for any power series g in two variables (and therefore for any polynomial in
two variables). We do not know whether ϕg is a p-fractal in general, but
we do know that the property of being a p-fractal is preserved under some
operations. This, combined with the results mentioned above, shows the
rationality of the Hilbert-Kunz series of a large family of polynomials.
Proposition 2.60. If ϕg is a p-fractal, then so is ϕgm .
Proof: Using the notation introduced before proposition 2.8, it’s clear that
(i)
(0)
ϕgm = ϕg while, for any 0 < i < m, ϕgm = 1. So the result follows from
proposition 2.8.
Proposition 2.61. Suppose f ∈ k[x1 , . . . , xs ] and g ∈ k[y1 , . . . , yt ]. If ϕf
and ϕg are p-fractals, then so is ϕf g .
56
Proof: Let I := (xq : f a ) and J := (y q : g a ). Then ((xq , y q ) : f a ) = (I, y q );
so ((xq , y q ) : (f g)a ) = (I, J). Then
deg(xq , y q , (f g)a ) = q s+t − deg(I, J)
= q s+t − deg(I) deg(J)
= q s+t − (q s − deg(xq , f a ))(q t − deg(y q , g a ))
= q t deg(xq , f a ) + q s deg(y q , g a ) − deg(xq , f a ) deg(y q , g a ),
and dividing by q s+t we get ϕf g ( aq ) = ϕf ( aq ) + ϕg ( aq ) − ϕf ( aq )ϕg ( aq ). By
continuity, ϕf g = ϕf + ϕg − ϕf ϕg , which gives the desired result.
Note that the previous propositions give an alternative proof of a result
proved by Chiang and Hung in [CH98]—the rationality of the Hilbert-Kunz
series of polynomials that can be expressed as a sum of monomials in disjoint
sets of variables. For ϕx is a p-fractal (trivially), and so is ϕg , for any
monomial g, by propositions 2.60 and 2.61. Hence any monomial is strongly
R-rational, and theorems 1.32 and 1.31 give the result.
In the remainder of this section, we’ll introduce some new families of
p-fractals involving polynomials in two variables. Once again, fix r ≥ 3,
and let `1 , . . . , `r be pairwise prime linear forms. A partition is a triple
S := (S1 , S2 , S3 ) of disjoint nonempty sets whose union is {1, . . . , r}. As in
section 2.4, ` = `1 `2 · · · `r and `a = `a11 `a22 · · · `ar r (a ∈ Nr ). Moreover, if A is
Y
Y
a subset of {1, 2, . . . , r}, then `A :=
`i and `aA :=
`ai i .
i∈A
57
i∈A
Definition 2.62. Let S = (S1 , S2 , S3 ) be a partition. Then δS and ϕS are
the continuous functions [0, ∞)r → [0, ∞) whose values at aq ∈ [0, ∞)rp are
given by
δS
a
:=
q
1
· δ(`aS1 , `aS2 , `aS3 )
q
and
ϕS
a
q
:=
1
· deg(`aS1 , `aS2 , `aS3 ).
q2
Theorem 2.63. The restriction of δS to [0, 1]r is a p-fractal.
Proof: Let q be a power of p and a ∈ [q]r . Let q := (q, q, . . . , q). Using
proposition 2.17 we obtain:
δS ( aq ) =
1
q
· δ(`aS1 , `aS2 , `aS3 )
=
1
q
q−a a
· δ(`qS1 , `qS2 , `q−a
S1 `S2 `S3 )
Hence δS ( aq ) = δa ◦ R( aq ), where a = (`S1 , `S2 ) and R : [0, 1]r → [0, 1]r is the
reflection given by:
R(x)i =



 xi
if i ∈ S3


 1 − xi if i 6∈ S3
Therefore δS = δa ◦R, by continuity, and the theorem follows from proposition
2.5 and theorem 2.49.
Corollary 2.64. The restriction of ϕS to [0, 1]r is a p-fractal.
58
Definition 2.65. Let S = (S1 , S2 , S3 ) be a partition, and let C = (F, G, H)
be a triple of nonzero homogeneous polynomials. Suppose that F `S1 , G`S2 ,
and H`S3 have no common factor. Then δS,C : [0, ∞)r → [0, ∞) and ϕS,C :
[0, ∞)r → [0, ∞) are the continuous functions whose values at aq ∈ [0, ∞)rp
are
δS,C
a
q
:=
1
· δ(F q `aS1 , Gq `aS2 , H q `aS3 )
q
and
ϕS,C
a
q
:=
1
· deg(F q `aS1 , Gq `aS2 , H q `aS3 ).
2
q
Theorem 2.66. The restriction of δS,C to [0, 1]r is a p-fractal.
Proof: Let q be a power of p and a ∈ [q]r . Let q := (q, q, . . . , q). Using
proposition 2.17 we obtain:
δS,C ( aq ) =
1
q
· δ(F q `aS1 , Gq `aS2 , H q `aS3 )
=
1
q
q−a a
· δ((F `S1 )q , (G`S2 )q , H q `q−a
S1 `S2 `S3 ).
Thus δS,C ( aq ) = δC1 ◦ R( aq ), where C1 = (F `S1 , G`S2 , H) and R is as in the
proof of theorem 2.63. Therefore δS,C = δC1 ◦ R, by continuity, and the
theorem follows from proposition 2.5 and theorem 2.49.
Corollary 2.67. The restriction of ϕS,C to [0, 1]r is a p-fractal.
Definition 2.68. Let C = (F, G, H) be a triple of nonzero, pairwise prime
homogeneous polynomials. Then δC∗ : [0, ∞)3 → [0, ∞) and ϕ∗C : [0, ∞)3 →
59
[0, ∞) are the continuous functions whose values at aq ∈ [0, ∞)3p are
δC∗
a
q
:=
1
· δ(F a1 , Ga2 , H a3 )
q
and
ϕ∗C
a
q
:=
1
· deg(F a1 , Ga2 , H a3 ).
q2
Theorem 2.69. The restriction of δC∗ to [0, 1]3 is a p-fractal.
Proof: By replacing k by a larger field, if necessary, we can assume that
F , G and H can be written as products of linear forms in k[x, y]. Explicitly,
if `1 , . . . , `r are all the linear factors of F , G and H, then we may write F =
Y
Y
Y
mi
i
i
`m
,
G
=
`
and
H
=
`m
i
i
i , for some partition S = (S1 , S2 , S3 ).
i∈S1
i∈S2
i∈S3
Let ψ be the continuous function [0, 1]r → [0, ∞) whose value at a/q ∈ [0, 1]rp
is
ψ
a
q
1
= ·δ
q
!
Y
i ai
`m
,
i
i∈S1
Y
i∈S2
i ai
`m
,
i
Y
i ai
`m
i
.
i∈S3
Using the notation introduced in the end of section 2.1, we clearly have
ψ (0) = δS , while ψ (i) is a function of the type δS,C1 , for all i 6= (0, . . . , 0). More
precisely, ψ (i) = δS,C1 for C1 = (`iS1 , `iS2 , `iS3 ). Thus proposition 2.8 shows that
ψ is a p-fractal. Let D be the map [0, 1]3 → [0, 1]r , (x1 , x2 , x3 ) 7→ (y1 , . . . , yr ),
where yi = xj if i ∈ Sj . Then δC∗ = ψ ◦D on [0, 1]3 , and the result now follows
from proposition 2.7.
Corollary 2.70. The restriction of ϕ∗C to [0, 1]3 is a p-fractal.
60
Chapter 3
Four Dimensional Fractals
Let k be a characteristic p field. Fix four pairwise prime linear forms `1 ,
Q
`2 , `3 , `4 ∈ k[x, y]. As in section 2.4, A := k[x, y]/( `i ), and A is the set
of homogeneous, zero-dimensional, non-principal ideals a = (F , G) of A. In
this chapter we will take a closer look at the four dimensional fractals δa (cf.
definition 2.32).
We will classify the elements of A up to the equivalence relation introduced in 2.40. We shall also see that the δa have interesting symmetry
properties. This will allow us to determine “blow-up rules” like those found
experimentally in example 2.28, that make possible explicit calculations such
as that of the Hilbert-Kunz series of polynomials of the type F (x, y)+G(z, t),
where F and G are homogeneous of degree 4.
61
3.1
Ideal Classes and the Fractals δa
We’ll start by looking at the simplest cases—when the degrees of the generators of the ideal a differ by at least two. We’ll show that if this is the
case, then there aren’t many possibilities for the fractal δa . Moreover, δa is
determined by its values at the corners of [0, 1]4 . That being done, we’ll then
look at the more complex and rich cases—the cases when the generators of
a have the same degree, and when the degrees of the generators differ by 1.
Before we start, we introduce the following notation: ε1 is the corner
(1, 0, 0, 0) of [0, 1]4 , and ε2 , ε3 and ε4 are defined likewise. Then εij := εi + εj ,
εijk := εi + εj + εk , and ε1234 := (1, 1, 1, 1).
3.1.1
Ideal Classes with δa (0) ≥ 4
Suppose δa (0) ≥ 4, meaning that a has generators F and G with deg F −
deg G ≥ 4. Then δa is linear, according to lemma 2.44. More precisely,
δa = deg F − deg G +
X
Xi −
i∈B
X
Xi ,
i6∈B
where B := {i | `i divides G}.
3.1.2
Ideal Classes with δa (0) = 3
If δa (0) = 3, suppose a = (F , G) with deg F − deg G = 3. If no `i divides G,
then F is a k-linear combination of x3 G, x2 yG, xy 2 G and y 3 G, and (F , G)
is principal, a contradiction. So δa equals 4 at one or more of the corners
62
εi , in which case δa |{0,1}4 takes on a maximum value ≥ 4, and theorem 2.38
shows that δa is linear on [0, 1]4 . Explicitly, if the maximum is attained at
the corner u, then δa (x) = δa (u) − d(x, u), for all x ∈ [0, 1]4 .
3.1.3
Ideal Classes with δa (0) = 2
If δa (0) = 2, then there are two possibilities:
A. If one of the `i , say `1 , divides G, then δa (ε1 ) = 3. So either δa |{0,1}4
takes on a maximal value ≥ 4 and δa is linear, or δa |{0,1}4 has local
maxima at ε1 and ε234 . Since for any x ∈ [0, 1]4 , either d(x, ε1 ) ≤ 3 or
d(x, ε234 ) ≤ 1, theorem 2.38 gives us:
δa (x) = max{3 − d(x, ε1 ), 1 − d(x, ε234 )}.
B. Suppose no `i divides G—so δa (εi ) = 1, for all i. We shall show that
there is actually only one ideal class with this property.
Take G of degree 2 not divisible by any `i , and F of degree 4 with F not
a linear combination of x2 G, xyG, and y 2 G—which is possible, since
Q
the subspace of degree 4 elements in k[x, y]/( `i ) is four dimensional.
Then (F , G) is such an ideal class.
As for the uniqueness, if b = (F1 , G1 ) is another ideal that falls into
this category, then we claim that a ∼ b. We’re free to multiply a
and b by polynomials prime to each `i , so we may assume that G1 =
63
Q
G. The space of elements of degree d = deg F in k[x, y]/( `i ) is
four dimensional, and therefore spanned by the linearly independent
elements x2 G, xyG, y 2 G and F . In particular, F1 can be written as a
linear combination of those, showing that a = b.
Note that δa (x) = 2 − d(x, 0) for any x with d(x, 0) ≤ 2, by theorem 2.38. In particular, δa (εij ) = 0, and therefore δa (εijk ) = 1. Now
δa (ε1234 ) must be 0—for if it were 2, then there would be a relation
between F , G and `1 `2 `3 `4 of degree d, implying that the ideal a is
principal. So this is the first ideal class we’ve found so far with the
property that δa is ≤ 1 at all corners except one. We give the ideal
classes with this property a special name:
Definition 3.1. An ideal class [a] is called quasi-regular if δa ≤ 1 at all
corners but one (where δa must equal 2).
We shall see later that there are exactly 16 quasi-regular ideal classes,
one for each corner, and that the associated δa are all “reflections” of one
another.
3.1.4
Ideal Classes with δa (0) = 0
The simplest cases being taken care of, we’ll now classify the ideals a = (F , G)
with deg F = deg G up to equivalence. Let A0 ⊂ A be the subset consisting
of such ideals.
64
Coordinatizing A0 —
For each a ∈ A0 we define two vectors in k4 :
u := (Res(F, `1 ), Res(F, `2 ), Res(F, `3 ), Res(F, `4 ))
and
v := (Res(G, `1 ), Res(G, `2 ), Res(G, `3 ), Res(G, `4 )).
Since the `i are linear, the resultants actually take a very simple form; namely,
if `i = ai x + bi y, then Res(F, `i ) = F (bi , −ai ). For simplicity, we’ll often use
F (i) to denote Res(F, `i ).
The assumption that a is non-principal guarantees not only that u and
v are nonzero vectors, but also that u and v are linearly independent. For
suppose u = λv for some scalar λ. Then all the `i would divide F − λG, and
the ideal a would be principal, contradicting our assumption. Thus u and v
span a two-dimensional subspace V (a) of k4 . In other words, to each ideal
a ∈ A0 we attach a k-rational point V (a) of the Grassmanian variety G(2, 4),
giving a mapping:
V : A0 −→ G(2, 4)
a
7−→
V (a)
Recall that G(2, 4) can be embedded as a quadric hypersurface in P5 in
the following way: if V ∈ G(2, 4) is spanned by vectors u and v, then V can
be viewed as the point of P5 with homogeneous coordinates vij := ui vj − uj vi
65
(1 ≤ i < j ≤ 4). Then G(2, 4) ⊂ P5 is the quadric
v12 v34 − v13 v24 + v14 v23 = 0.
(3.1)
This embedding is known as the Plucker embedding, and the coordinates vij
called Plucker coordinates. We’ll often talk about the coordinates of an ideal
a, by which we mean the Plucker coordinates of V (a).
Note that, because of our assumptions on a, `i cannot divide both F and
G. Thus V (a) is not contained in any of the hyperplanes xi = 0.
Definition 3.2. X is the subset of G(2, 4) which consists of all the k-rational
subspaces V contained in none of the hyperplanes xi = 0.
We’ll now verify how the vanishing of a coordinate vij of a = (F , G)
relates to the function δa . If vij = 0, then F (i) G(j) − F (j) G(i) = 0. So there
exist a, b ∈ k, not both zero, such that aF (i) + bG(i) = aF (j) + bG(j) = 0.
But this means that both `i and `j divide aF + bG, and therefore there
exists a relation between F , G and `i `j of degree equal to the degree of F
and G. Hence δ(F, G, `i `j ) > 0. Actually δ(F, G, `i `j ) = 2, to be more
precise, given proposition 2.19 and the fact that δ(F, G, 1) = 0. Conversely,
if δ(F, G, `i `j ) > 0, then vij = 0, establishing the following result:
Proposition 3.3. A coordinate vij of a is zero if and only if δa (εij ) = 2. 66
The Torus Action—
The four dimensional torus T := (k∗ )4 acts on k4 by coordinatewise multiplication, which induces an action of T on G(2, 4). In coordinates, this action
can be described as follows: if t = (t1 , . . . , t4 ) ∈ T and V = (vij ) ∈ G(2, 4),
then t · V is the point with coordinates ti tj vij .
Now suppose U is a homogeneous polynomial prime to each `i . Then
V (U a) is spanned by (U (1) F (1) , . . . , U (4) F (4) ) and (U (1) G(1) , . . . , U (4) G(4) ),
and hence V (U a) = (U (1) , . . . , U (4) ) · V (a). Thus, if a ∼ b, then V (a) and
V (b) lie in the same torus orbit, and we have a well-defined mapping
V : A0 /∼ −→
[a]
X/T
7−→ T · V (a)
where [a] denotes the equivalence class of a.
Proposition 3.4. V (a) and V (b) lie in the same orbit if and only if a ∼ b;
i.e. the map V : A0 /∼ −→ X/T is bijective.
Proof: Surjectivity is clear: if W ∈ X is spanned by u = (u1 , . . . , u4 ) and
v = (v1 , . . . , v4 ), then one can find forms F and G of the same degree with
F (i) = ui and G(i) = vi , for i = 1, 2, 3, 4. The fact that W is not contained
in any of the hyperplanes xi = 0 implies that no `i divides both F and G,
while the fact that u and v are linearly independent shows that a = (F , G)
is not principal. So a ∈ A0 , and V (a) = W .
67
As for the the injectivity of V , suppose a = (F , G), b = (P , Q), and
V (b) = (u1 , u2 , u3 , u4 ) · V (a). We’re free to multiply a and b by polynomials
prime to the `i , so we may assume that the degrees of F , G, P and Q are all
the same. Then we can write
(P (1) , . . . , P (4) ) = a(u1 F (1) , . . . , u4 F (4) ) + b(u1 G(1) , . . . , u4 G(4) )
(3.2)
(Q(1) , . . . , Q(4) ) = c(u1 F (1) , . . . , u4 F (4) ) + d(u1 G(1) , . . . , u4 G(4) ),
(3.3)
and
where det
ab
cd
6= 0.
Now let U and H be homogeneous polynomials of the same degree such
that Res(U, `i ) = ui and Res(H, `i ) = 1 for all i (one can find such polyQ
nomials of degree 3). Then (3.2) and (3.3) show that
`i divides both
HP − U (aF + bG) and HQ − U (cF + dG). So
b ∼ Hb = U aF + bG, cF + dG
= Ua
∼ a,
giving the desired result.
Corollary 3.5. If V ([a]) = V ([b]), then δa = δb .
68
Classification of the Ideal Classes in A0 —
From the above proposition, the problem of classifying ideal classes in A0
reduces naturally to the problem of classifying the torus orbits in X. Let
T · V ∈ X/T . There are three possible situations:
Case 1. V contains a vector with exactly 3 coordinates equal to 0. Without
loss of generality, we can suppose (0, 0, 0, 1) ∈ V . Then V also contains a
vector (a, b, c, 0), and since V is not contained in any hyperplane xi = 0, a,
b and c must be nonzero. Acting by the torus, we may assume that V is
spanned by (0, 0, 0, 1) and (1, 1, 1, 0).
The coordinates v12 , v13 and v23 of V are zero, while the other coordinates
are nonzero. So, if V = V ([a]) then δa (ε12 ) = δa (ε13 ) = δa (ε23 ) = 2, and δa is
zero at the other corners εij . Furthermore, since δa (ε1 ) = 1, proposition 2.23
shows that δa (ε123 ) = 3. Now we can easily see that δa |{0,1}4 attains local
maxima at ε123 and ε4 , and we are in a situation similar to 3.1.3-A. Thus,
δa (x) = max{3 − d(x, ε123 ), 1 − d(x, ε4 )}.
So δa is piecewise linear, and completely determined by the coordinates of a
that vanish.
Case 2. V contains a vector with precisely 2 coordinates equal to zero. So we
may assume, without loss of generality, that (0, 0, e, f ) ∈ V , where e, f 6= 0.
We may assume e = f = 1. Then V also contains a vector (a, b, 0, c) with
a, b 6= 0. If c = 0, then acting by the torus we may assume that a = b = 1,
69
and V is spanned by (0, 0, 1, 1) and (1, 1, 0, 0). If c 6= 0, then acting by the
torus we may assume a = c and b = c. Then V is spanned by (0, 0, 1, 1) and
(1, 1, 0, 1).
In the first case (c = 0) the coordinates v12 , and v34 of V are zero, while the
other coordinates are nonzero. So, if V = V ([a]) then δa (ε12 ) = δa (ε34 ) = 2,
and δa vanishes at all the other corners εij . This and proposition 2.19 imply
that δa is 1 at all the corners εijk . Moreover, since δa (0) = 0, δa (εi ) = 1 for
all i. So δa |{0,1}4 attains local maxima at ε12 and ε34 , and we can use theorem
2.38, obtaining:
δa (x) = 2 − min{d(x, ε12 ), d(x, ε34 )}
= max{2 − d(x, ε12 ), 2 − d(x, ε34 )}.
Again, δa is piecewise linear, and completely determined by the coordinates
of a that vanish.
In the second case (c 6= 0) only v12 is zero. If V = V ([a]), then δa (ε12 ) = 2,
and δa vanishes at all the other corners εij . Then it’s easy to see that δa is
≤ 1 at all the other corners—so [a] is quasi-regular. There are obviously six
torus orbits of this type (one for each coordinate vij ), and there are six ideals
classes [aij ] mapping to these orbits. These classes are represented by the
70
following ideals:
2
a12 = (`1 `2 , `3 )
2
a34 = (`3 `4 , `1 )
2
a13 = (`1 `3 , `2 )
2
a24 = (`2 `4 , `1 )
2
a14 = (`1 `4 , `3 )
2
a23 = (`2 `3 , `1 )
Case 3. No element of V has more than one coordinate equal to zero. Then
V contains the vectors (0, 1, a, b) and (1, 0, c, d), with a, b, c, d 6= 0. Acting
by the torus, we may assume a = b = 1. Acting by the torus again, we
may assume that V is spanned by (0, 1, 1, 1) and (c, 0, c, d) or, simpler yet,
(0, 1, 1, 1) and (1, 0, 1, λ), where λ 6= 0, 1.
Let X0 ⊂ X consist of all the subspaces V of this type. X0 can also be
thought of as the set of all V with no Plucker coordinate equal to 0. The
function
X0
−→
k
12 v34
V = (vij ) 7−→ vv13
v24
is constant on the orbits, therefore inducing a mapping X0 /T −→ k. Note
71
that this function maps the subspace V above, spanned by (0, 1, 1, 1) and
(1, 0, 1, λ), to 1 − 1/λ, and therefore it maps X0 /T bijectively onto k \ {0, 1}.
So if |k| = q is finite, there are q − 2 orbits of this type.
12 v34 .
Definition 3.6. γ : X0 /T → k \ {0, 1} is the bijection T · V 7−→ vv13
v24
Suppose V = V ([a]) ∈ X0 /T . By abuse of notation, we’ll write γ(a)
for γ(V ([a])). Then the ideal class [a] (and consequently δa ) is completely
determined by its parameter or invariant t := γ(a), and there are q − 2 such
ideal classes. After a linear change of variables we may assume that `1 = x,
`2 = y, `3 = x + y and `4 = x + cy, for some c 6= 0, 1, and then we can
explicitly find an ideal at such that γ(at ) = t, for each t ∈ k \ {0, 1}. In fact,
we may take
at = (c + t − 1)x2 + (c2 + t − 1)xy, y 2
if t 6= 1 − c, or at = (x, y) if t = 1 − c.
If V ([a]) ∈ X0 /T , then none of the coordinates vij of a is zero, and δa
vanishes at all the corners εij . So, unless δa (ε1234 ) = 2, δa will be less than
or equal to 1 at all corners. We give these ideal classes a special name:
Definition 3.7. An ideal class [a] is called regular if δa ≤ 1 at all corners
of [0, 1]4 .
Definition 3.8. An ideal class [a] is called even (resp. odd) if δa (0) is even
(resp. odd).
We can now state and prove the main theorem of this section:
72
Theorem 3.9. If q := |k|, then there are precisely q − 3 regular even ideal
classes, and 8 quasi-regular even ideal classes.
Proof: We’ve already seen that there are 7 quasi-regular ideal classes [a]:
one with δa (0) = 2 and, for each corner εij , one with δa (εij ) = 2. We’ve also
seen that there are q − 2 “parameterized” ideal classes with no coordinate
equal to zero. Thus, in view of the remark made before definition 3.7, it only
remains to be proven that within the latter group there is precisely one ideal
class [a] with δa (ε1234 ) = 2.
The existence of this ideal class is clear: the class represented by (x, y)
has the required properties. In the next section, using reflections, we shall
give a more general uniqueness result, valid not only for four linear forms
`1 , `2 , `3 , `4 , but for any number of forms. But since the proof in the case at
hand is straightforward, we present it here.
Suppose a = (F , G), with deg F = deg G = d, and suppose [a] is quasiregular, with δa (ε1234 ) = 2. The fact that δ(F, G, `1 `2 `3 `4 ) = 2 shows that
there is a relation of degree d + 1 between F , G and `1 `2 `3 `4 . Equivalently,
there are a, b, c, d ∈ k, not all equal to zero, such that all the `i divide
(ax−by)F +(cx−dy)G—or, in other words, Res((ax−by)F +(cx−dy)G, `i ) =
0. Now if we write `i = ai x + bi y, this is equivalent to saying that there is a
73
non trivial solution (a, b, c, d) to the following system:




b1 F (1) a + a1 F (1) b + b1 G(1) c + a1 G(1) d = 0







 b2 F (2) a + a2 F (2) b + b2 G(2) c + a2 G(2) d = 0



b3 F (3) a + a3 F (3) b + b3 G(3) c + a3 G(3) d = 0







 b4 F (4) a + a4 F (4) b + b4 G(4) c + a4 G(4) d = 0
The determinant of this system is
Res(`1 , `2 )Res(`3 , `4 )v13 v24 − Res(`1 , `3 )Res(`2 , `4 )v12 v34 ,
so the existence of a nontrivial solution shows that [a] must be the ideal class
Res(`1 , `2 )Res(`3 , `4 )
with parameter t = γ(a) =
. Note that when `1 = x,
Res(`1 , `3 )Res(`2 , `4 )
`2 = y, `3 = x + y and `4 = x + cy this quasi-regular class has invariant
t = 1 − c.
Alternatively, we may assume `1 = x, `2 = y, `3 = x + y and `4 = x + cy.
If t 6= 1 − c, let F = (c + t − 1)x2 + (c2 + t − 1)xy and G = y 2 . Then F and
G have no common factor, and a direct calculation shows that each vij 6= 0
and (F , G) has invariant t. Since deg F = deg G = 2, the corresponding δ is
0 at ε1234 , and (F , G) is a regular class with invariant t.
3.1.5
Interlude: Reflections
The analysis of the remaining case, the ideal classes with δa (0) = 1, depends on our setting up a correspondence between these ideal classes and
74
those already studied, in such a way that the corresponding functions δa are
“reflections” of one another. In this section we’ll develop the theory of reflections in the most general situation, in which we have r pairwise prime linear
forms `1 , . . . , `r .
To make these ideas precise, we’ll have to define the aforementioned correspondence not exactly on the set of ideal classes, but on a larger set, by
introducing a new notion of equivalence:
Definition 3.10. Consider the set of all (unordered) pairs of nonzero forms
{F, G} such that no `i divides both F and G. On this set we introduce the
equivalence relation ∼ generated by the following rules:
• {F, G} ∼ {aF, bG}, for any a, b ∈ k∗ .
• {F, G} ∼ {F, G∗ } if G ≡ G∗ (mod F ).
• {F, G} ∼ {F ∗ , G∗ } if F ≡ F ∗ (mod
Q
`i ) and G ≡ G∗ (mod
Q
`i ).
• {F, G} ∼ {U F, U G} if U is a form prime to each `i .
We denote by X the set of equivalence classes of such pairs.
One easily verifies the following:
Proposition 3.11. There is an injection (A/∼) ,→ X that maps the class
of a non-principal ideal (F , G) to the class of the pair {F, G}.
75
In what follows we’ll identify A/∼ with its image in X via the map of
proposition 3.11. Now suppose the equivalence class I of {F, G} is not in
A/ ∼. Then a = (F , G) is a principal ideal. We attach an integer m to
I as follows. If deg F = deg G, then m := r. If deg F =
6 deg G, then
Q
Q
either F or G is divisible by
`i . We may assume that
`i divides F ;
then m := r + deg G − deg F . m doesn’t depend on the representative pair
Q
{F, G}. Furthermore, m determines the equivalence class I. For say
`i
divides both F and F ∗ and that deg G∗ − deg F ∗ = deg G − deg F . Then
{F, G} and {F ∗ , G∗ } are both equivalent to {F G∗ , GG∗ }, which gives the
result when m 6= r. The case m = r is almost equally easy. We have shown
that X is the union of A/∼ with an explicit collection of classes, one for each
m ∈ Z.
Definition 3.12. Suppose I ∈ X is the equivalence class of the pair {F, G}.
Then δI is the continuous function [0, 1]r −→ [0, ∞) whose value at aq ∈
[0, 1]rp is
δI
a
q
=
1 q q Y ai · δ F ,G ,
`i .
q
Note that δI depends only on the equivalence class I, and not on the
choice of the representative. Also note that either a = (F , G) is non-principal,
and δI = δa , or a is principal, and δI is a piecewise linear function of the type
P
|m − Xi |, for some m ∈ Z (cf. remark made in the beginning of section
2.4).
Let R1 be the reflection in the first coordinate, i.e. the map [0, 1]r −→
76
[0, 1]r , (x1 , . . . , xr ) 7−→ (1 − x1 , x2 , . . . , xr ). We’ll now show that there is a
bijection X → X , I 7→ J , such that δJ = δI ◦ R1 .
Lemma 3.13. Suppose {F, `1 G} and {H, `1 G} are as in definition 3.10.
Suppose moreover that deg F = deg H and F ≡ H (mod `2 . . . `r ). Then
{F, `1 G} ∼ {H, `1 G}.
Proof:
Choose forms U and V of the same degree with U (1) = H (1) ,
V (1) = F (1) , and U (i) = V (i) = 1, for i > 1. Then, since F (i) = H (i) for i > 1,
Q
(U F )(i) = (V H)(i) for all i, and U F ≡ V H (mod
`i ). Also (U `1 )(i) =
Q
(V `1 )(i) for all i, and so U `1 G ≡ V `1 G (mod
`i ). Thus {U F, U `1 G} ∼
{V H, V `1 G}, giving the result.
Now let I ∈ X , and choose a representative {F, G} of I with deg F ≥ r
and deg G ≥ r. Modifying one of these elements by a multiple of the other,
Q
we may assume that F = `1 F ∗ . Modifying F by a multiple of `i we may
assume that `1 doesn’t divide F ∗ . Since deg G ≥ r, G is congruent to some
`1 G∗ (mod `2 · · · `r ), with G∗ 6= 0. Now let J be the class of {F ∗ , `1 G∗ }.
Lemma 3.14. J depends only on I, and not on the choice of F , G, F ∗ or
G∗ .
First fix F and G, with F = `1 F ∗ . Then if `1 G∗ and `1 G#
Q
are both congruent to G (mod `2 · · · `r ), then `1 G∗ ≡ `1 G# (mod
`i ),
Proof:
so {F ∗ , `1 G∗ } ∼ {F ∗ , `1 G# }.
77
`i . Then F ∗ is
Q
modified by a multiple of `2 · · · `r and `1 G∗ is modified by a multiple of `i ,
Suppose now we modify F and G by multiples of
Q
so the result follows from lemma 3.13.
Next suppose we replace F by F +AG = `1 F # , where `1 6 |F # . Then since
`1 divides F , `1 divides A, and therefore F # = F ∗ + BG, for some B. Then
F # ≡ F ∗ (mod `1 G∗ , `2 · · · `r ), and lemma 3.13 shows that {F ∗ , `1 G∗ } ∼
{F # , `1 G∗ }.
Now if we replace G by G + AF , then we replace G∗ by G∗ + AF ∗ , and
{F ∗ , `1 G∗ } is replaced by the equivalent pair {F ∗ , `1 (G∗ + AF ∗ )}.
Finally, suppose we replace F and G by U F and U G, where U is a form
prime to each `i . Then we may replace F ∗ and G∗ by U F ∗ and U G∗ , so that
{F ∗ , `1 G∗ } is replaced by {U F ∗ , U `1 G∗ }, an equivalent pair. The same goes
if we multiply F and G by nonzero scalars.
Definition 3.15. The reflection R1 is the map I 7−→ J (by abuse of
notation).
Note:
The reason we were forced to “extend” the set of ideal classes in
order to define the reflection mapping R1 was because A/ ∼ is not stable
under R1 . For instance, R1 maps the class of the ideal a = (`1 , `2 · · · `r ) to
an element in X \ (A/∼).
Theorem 3.16. R1 ◦ R1 is the identity map.
78
Proof: From an equivalence class I we take a representative {`1 F ∗ , G},
with deg `1 F ∗ ≥ r and deg G ≥ r, and `1 6 |F ∗ . Then we choose G∗ such that
`1 G∗ ≡ G (mod `2 · · · `r ), and clearly G∗ can also be chosen so that `1 6 |G∗ .
Then R1 (I) is the class J represented by {F ∗ , `1 G∗ }.
Now choose V of degree r − 2 not divisible by any `i , and let U =
V `1 + `2 · · · `r . Then J is also represented by {U F ∗ , `1 U G∗ }. Since U ≡
V `1 (mod `2 · · · `r ), U F ∗ ≡ V `1 F ∗ (mod `2 · · · `r ) and R1 (J ) is the class of
{V `1 F ∗ , U G∗ }. Now since U G∗ ≡ V `1 G∗ ≡ V G (mod `2 · · · `r ), lemma 3.13
shows that R1 (J ) is the class of {V `1 F ∗ , V G}. Thus R1 (J ) = I, proving
the theorem.
Theorem 3.17. δR1 (I) = δI ◦ R1 .
Proof: Choose a representative {`1 F ∗ , G} for I, with `1 6 |F ∗ , and suppose
`1 G∗ ≡ G (mod `2 · · · `r ). Then R1 (I) is the equivalence class represented
by {F ∗ , `1 G∗ }. Now for any q and any a ∈ [q]r ,
δI
a
q
1 q ∗ q q Y ai · δ `1 F , G ,
`i
q
r
1 q−a1 ∗ q q Y ai =
· δ `1 F , G ,
`i
q
i=2
r
Y
1
q−a1 ∗ q q ∗ q
=
· δ `1 F , `1 G ,
`ai i
q
i=2
r
Y 1
∗ q a1 ∗ q
=
· δ F , `1 G ,
`ai i
q
i=2
r
Y
1
ai
1
=
· δ F ∗ q , `q1 G∗ q , `q−a
`
.
1
i
q
i=2
=
79
Thus δI ( aq ) = δR1 (I) ◦ R1 ( aq ) for all
a
q
∈ [0, 1]rp , and the result follows by
continuity.
Just as we defined R1 : X −→ X , we may define Ri : X −→ X for
1 ≤ i ≤ r and prove the analogues of the theorems above. The Ri may be
shown to commute with one another, and so we get an action of (Z/2)r on
X . Call an element I of X special if δI = r − 2 at one corner of [0, 1]r ,
δI = r − 3 at all adjacent corners and δI = 0 at the opposite corner. (When
r = 4, special is the same as quasi-regular; see definition 3.1.) The set of
special elements is stable under the action of (Z/2)r . In fact, (Z/2)r acts
transitively on the special elements:
Theorem 3.18. Any two special elements I and J are in the same orbit of
(Z/2)r . Consequently the functions δI and δJ are reflections of one another.
Proof: It suffices to show that there is only one special equivalence class
I with δI (0) = r − 2. This may be proved by an argument similar to
that given in 3.1.3. Let {F, G} be a representative pair for I, and suppose
deg F −deg G = r−2. Then no `i divides G. Now suppose {F1 , G1 } is another
pair with the same property. As in 3.1.3, we may assume that G1 = G. The
Q
space of elements of degree d = deg F in k[x, y]/( `i ) is r dimensional, and
therefore spanned by the linearly independent elements xr−2 G, xr−3 yG,. . . ,
y r−2 G and F . In particular, F1 can be written as a linear combination of
these elements, showing that {F, G} ∼ {F1 , G}.
80
3.1.6
Back to Four Dimensions
Theorem 3.19. R1 maps the regular (resp. quasi-regular) odd ideal classes
bijectively onto the regular (resp. quasi-regular) even ideal classes. So if
R1 ([a]) = [a∗ ], then δa∗ = δa ◦ R1 .
Proof: Suppose [a] is a regular odd ideal class, and let J = R1 ([a]). Then
theorem 3.17 shows that δJ is ≤ 1 at all corners of [0, 1]r . In particular, δJ
P
cannot be a function of the type |m − Xi |, and therefore J must lie in
A/ ∼. So R1 ([a]) is a regular even ideal class. Likewise, if [a] is a regular even
ideal class then R1 ([a]) is a regular odd ideal class and, since R1 ◦ R1 is the
identity map, the result follows. A similar argument is used for quasi-regular
ideal classes.
In view of theorems 3.9 and 3.19 we obtain:
Theorem 3.20. If |k| = q is finite, then there are exactly q − 3 regular odd
ideal classes, and 8 quasi-regular odd ideal classes.
The “interesting” ideal classes with δa (0) = 1—the regular odd ideal
classes—can be parameterized by using the reflection mapping R1 and the
function γ: to each regular odd ideal class [a] we assign a parameter t :=
γ(R1 ([a])). Any such [a] is therefore completely determined by its parameter
t.
We’ll conclude this section with the proof of the existence of certain
“canonical representatives” for the regular odd ideal classes.
81
Theorem 3.21. Any regular odd ideal class has a representative of the form
(P2 , `), where ` is a linear form prime to each of `1 , . . . , `4 , and deg P2 = 2.
Moreover, the ideal (P2 , `) only depends on `, and not on P2 .
Proof: Suppose a regular odd ideal class is represented by (F , G), with
deg F = deg G + 1. The images of the five forms xF , yF , x2 G, xyG and
Q
y 2 G in k[x, y]/( `i ) are linearly dependent, so `F ≡ P2 G (mod `1 `2 `3 `4 ),
for some ` and P2 .
Suppose ` = c`1 , for some c ∈ k. Since `1 doesn’t divide G (by regularity),
`1 divides P2 , and cF ≡ (P2 /`1 )G (mod `2 `3 `4 ). If c = 0, G is divisible by
`2 , `3 or `4 , contradicting regularity. If c 6= 0, then (F , G) = (`2 `3 `4 A, G)
for some A, and the function δ attached to this ideal is 2 at the corner ε234 ,
again contradicting regularity. So ` is not a multiple of `1 . Similarly, ` is
prime to `2 , `3 and `4 . Thus (F , G) ∼ (`F , `G) = (P2 G, `G) ∼ (P2 , `).
To conclude the proof, note that if m = (x, y), then (P2 , `) = (m2 , `),
showing that (P2 , `) only depends on `.
Note that the previous theorem gives another proof of the existence of
q − 3 regular odd ideal classes if |k| = q is finite!
3.1.7
Summary
We summarize the main results above for finite k in the following theorem:
Theorem 3.22. Let q := |k|. Then:
82
• There are q−3 regular even ideal classes [a], each of which is completely
determined by its parameter t = γ(a). If `1 = x, `2 = y, `3 = x + y
and `4 = x + cy, then t takes all values except 0, 1 and 1 − c, and [a]
is represented by
at = (c + t −
1)x2
+
(c2
+t−
1)xy, y 2
.
• There are q − 3 regular odd ideal classes.
• There exists a bijection R1 : [a] 7−→ [a∗ ] between regular (resp. quasiregular) odd ideal classes and regular (resp. quasi-regular) even ideal
classes, in such a way that δa∗ = δa ◦ R1 , where R1 (x1 , x2 , x3 , x4 ) :=
(1 − x1 , x2 , x3 , x4 ).
• There are 16 quasi-regular ideal classes. For each corner of [0, 1]4 , there
is exactly one quasi-regular ideal class [a] such that δa equals 2 at that
corner. The functions δ attached to the quasi-regular ideal classes are
all reflections of one another.
• For all the other ideal classes [a], δa is either linear or piecewise linear.
3.2
Symmetry Theorems
In this section we’ll show that the fractals attached to regular ideal classes
bear some remarkable symmetry properties.
83
Definition 3.23. R : X −→ X is the map R1 ◦ R2 ◦ R3 ◦ R4 . We also denote
by R the reflection (x1 , . . . , x4 ) 7−→ (1 − x1 , . . . , 1 − x4 ).
Our first result shows that regular ideal classes are invariant under R.
Lemma 3.24. If [a] is a regular odd ideal class, then the products γ(R1 ([a]))·
γ(R4 ([a])) and γ(R2 ([a]))·γ(R3 ([a])) are equal, and do not depend on the ideal
class [a]. More precisely,
γ(R1 ([a])) · γ(R4 ([a])) = γ(R2 ([a])) · γ(R3 ([a])) =
Res(`1 , `2 ) · Res(`3 , `4 )
.
Res(`1 , `3 ) · Res(`2 , `4 )
Proof: We may assume a = (`1 `4 , `), with ` as in theorem 3.21. Let U
be a form of degree ≥ 3 with Res(U, `i ) = 1, for all i. Now choose F ≡ U `
(mod `2 `3 `4 ) such that `1 |F , and G ≡ U ` (mod `1 `2 `3 ) with `4 |G. Then
R1 ([a]) is the class of (U `4 , F ), and R4 ([a]) is the class of (U `1 , G).
Our choice of F assures that Res(F, `1 ) = 0, while Res(F, `i ) = Res(`, `i ),
for i = 2, 3, 4. So V (R1 ([a])) is spanned by the vectors
(Res(`4 , `1 ), Res(`4 , `2 ), Res(`4 , `3 ), 0)
and
(0, Res(`, `2 ), Res(`, `3 ), Res(`, `4 )),
and therefore γ(R1 ([a])) =
γ(R4 ([a])) =
Res(`3 ,`4 )Res(`,`2 )
.
Res(`2 ,`4 )Res(`,`3 )
Res(`1 ,`2 )Res(`,`3 )
,
Res(`1 ,`3 )Res(`,`2 )
A similar calculation shows that
so
γ(R1 ([a])) · γ(R4 ([a])) =
Res(`1 , `2 ) · Res(`3 , `4 )
.
Res(`1 , `3 ) · Res(`2 , `4 )
84
An identical argument gives the other identity.
Now the symmetry theorem follows easily:
Theorem 3.25. If [a] is a regular ideal class, then [a] is invariant under R.
So δa is invariant under R, i.e. δa (x1 , . . . , x4 ) = δa (1 − x1 , . . . , 1 − x4 ), for
all (x1 , . . . , x4 ) ∈ [0, 1]4 .
Proof: If the ideal class [a] is even, then the previous lemma applied to
R1 ([a]) and R2 ([a]) shows that γ([a])γ(R1 ◦ R4 ([a])) = γ([a])γ(R2 ◦ R3 ([a])),
and therefore R1 ◦ R4 ([a]) = R2 ◦ R3 ([a]), giving the result.
If [a] is odd, then it is the reflection of a regular even ideal class [b] under
R1 , and [a] = R1 ([b]) = R1 ◦ R([b]) = R ◦ R1 ([b]) = R([a]).
A simple but interesting symmetry result is the following:
Theorem 3.26. If [a] is a regular even ideal class, then δa (x1 , x2 , x3 , x4 ) =
δa (x2 , x1 , x4 , x3 ), for all (x1 , x2 , x3 , x4 ) ∈ [0, 1]4 . A similar identity holds when
one interchanges other pairs of variables xi ↔ xj , xk ↔ xl , with {i, j, k, l} =
{1, 2, 3, 4}.
Proof: We may assume that `1 = x, `2 = y, `3 = x + y, and `4 = x + cy.
Suppose a = (F , G). The linear change of variables x 7→ cy, y 7→ x maps
F 7→ F ∗ = F (cy, x) and G 7→ G∗ = G(cy, x), and essentially swaps `1 ↔ `2 ,
`3 ↔ `4 . (More precisely, `1 7→ c`2 , `2 7→ `1 , `3 7→ `4 , and `4 7→ c`3 .) So
δ(F ∗ q , G∗ q , `a12 `a21 `a34 `a43 ) = δ(F q , Gq , `a11 `a22 `a33 `a44 ),
85
for any q and any a ∈ [q]4 . Thus if a∗ = (F ∗ , G∗ ), then δa∗ (x2 , x1 , x4 , x3 ) =
δa (x1 , x2 , x3 , x4 ), for all (x1 , x2 , x3 , x4 ) ∈ [0, 1]4 . But the ideals a and a∗ are
equivalent—for one readily verifies that
(F ∗ (1) , F ∗ (2) , F ∗ (3) , F ∗ (4) ) = ((−c)d F (2) , (−1)d F (1) , (−1)d F (4) , (−c)d F (3) )
and
(G∗ (1) , G∗ (2) , G∗ (3) , G∗ (4) ) = ((−c)d G(2) , (−1)d G(1) , (−1)d G(4) , (−c)d G(3) ),
where d is the degree of F and G, and hence a and a∗ have the same invariant.
3.3
Blow-Up Rules
For the time being we drop the assumption that we are dealing with only
four linear forms `1 , `2 , `3 , `4 . We introduce an action of the semigroup T on
the set X . Suppose Tq|a ∈ T and {F, G} represents an element I of X . (In
particular, I could be the class of an ideal a ∈ A.) Two possibilities arise:
Q
Q
A. The image in k[x, y]/( `i ) of the colon ideal ((F q , Gq ) : `ai i ) is nonprincipal. Then this image corresponds to an element of X (more
precisely, an element of A/∼), immediately seen to be independent of
the representative {F, G} of I. Denote this element by Tq|a I. The
proof of theorem 2.49 shows that Tq|a δI = 1q δTq|a I .
86
B. The image of the colon ideal is principal. Then the proof of theorem
P
2.49, combined with lemma 2.48, shows that Tq|a δI = 1q |m −
Xi |,
for some integer m. There is a unique element J of X with δJ =
P
|m − Xi |, and we set Tq|a I equal to J .
The construction above gives an action of T on X . Furthermore, Tq|a δI =
1
δ
,
q Tq|a I
for all I in X .
Example 3.27. In example 2.28 we observed certain “rules” concerning the
transforms of the fractal δH , where p = 3, H = xy(x + y)(x + y), and
2 + 2 + 2 = 0.
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
δH (x) (0 ≤ x ≤ 1)
We showed that δH is linear on [ 12 , 1], so T3|2 δH is a linear function L. It
appeared that
(a) T3|1 δH = 13 δH ;
87
(b) T9|0 δH = 19 δH ;
(c) T9|2 δH = 19 δH ◦ R, where R is the reflection map t 7→ 1 − t;
(d) T9|1 δH = 91 |4X − 2|.
These rules can be summarized in the following self-explanatory diagram:
δH
A
A
7−→
7−→
1
δ
9 H
1
δ
3 H
1
|4X
9
L
− 2|
1
δ
9 H
◦R
To verify these “blow-up rules” we set `1 = x, `2 = y, `3 = x + y and
`4 = x + y, and look at the four dimensional fractal δa , where a = (x, y).
[a] is the only quasi-regular ideal class whose attached fractal δ is 2 at the
corner ε1234 , and δH (z) = δa (z, z, z, z).
We verify (a) by looking at T3|1 [a], where 1 := (1, 1, 1, 1). A direct calculation of the values of δT3|1 [a] at the corners of [0, 1]4 shows that T3|1 [a]
is quasi-regular, and δT3|1 [a] (ε1234 ) = 2, so T3|1 [a] = [a]. The same applies
to T9|0 [a], verifying (b). T9|2 [a], on the other hand, is quasi-regular with
δT9|2 [a] (0) = 2. Therefore δT9|2 [a] = δa ◦ R1234 , where R1234 is the reflection
(x1 , . . . , x4 ) 7−→ (1 − x1 , . . . , 1 − x4 ), and (c) follows. Finally, the function δ
attached to T9|1 [a] is 2 at 0 and at ε1234 , and ≤ 1 at the other corners. So it
88
is the piecewise linear function x 7−→ max{2 − d(x, 0), 2 − d(x, ε1234 )}. This
gives (d).
Example 3.28. We’ll now revisit example 2.30, and determine the blow-up
rules for that two dimensional “slice” of a four dimensional fractal. (The
method we use here is designed to illustrate the ideas and results of this
chapter, and is not styled for efficiency.) Here again p = 3, and `1 , `2 , `3 and
`4 are as in the previous example. Let k := Z/3(). Also, let a = (x, y), and
let δ be the function (z, w) 7−→ δa (z, z, w, w). Then the following graph shows
j
the values of the function (i, j) 7−→ 81 · δ( 81i , 81
) on the region i + j ≤ 81.
(Recall that the zeros of the function were replaced by dots, and on the
omitted region the function is linear.)
89
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80
We know that T3|(2,1) δ = T3|(1,2) δ = L1 and T3|(2,2) δ = L2 , where L1 and
L2 are linear functions. Furthermore, the graph suggests the following:
(a) T3|(2,0) δ = T3|(1,1) δ = T3|(0,2) δ = 13 δ;
(b) T3|(1,0) δ = T3|(0,1) δ =: α.
Let β := T3|(0,0) δ. Then these blow-up rules can be summarized in the
90
following diagram:
δ
7−→
1
δ
3
L1
L2
α
1
δ
3
L1
β
α
1
δ
3
In order to verify (a), we apply the transformations T3|(2,2,0,0) , T3|(1,1,1,1)
and T3|(0,0,2,2) to [a], and check that they all yield the quasi-regular class whose
attached δ is 2 at ε1234 —namely, [a]. As for (b), T3|(1,1,0,0) [a] = [(x2 , y 2 )],
which is the regular even ideal class with invariant t = 5 , and one verifies
that T3|(0,0,1,1) [a] also gives the same class.
j
The following graph shows the function (i, j) 7−→ 243 · α( 81i , 81
):
91
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. . . 2 . . . . . . . 2 . . . . .
. . 2 4 2 . . . . . 2 . . . . . .
. 2 4 6 4 2 . . . . . . . . . . .
2 4 6 8 6 4 2 . . . . . . . . . .
2 4 6 8 10 8 6 4 2 . . . . . . . . .
4 6 8 101210 8 6 4 2 . . . . . . . .
6 8 1012141210 8 6 4 2 . . . 2 . . .
6 8 10121416141210 8 6 4 2 . . . 2 . .
8 101214161816141210 8 6 4 2 . . . . .
6 8 10121416141210 8 6 4 2 . 2 . . . .
4 6 8 1012141210 8 6 4 2 . 2 4 2 . . .
2 4 6 8 101210 8 6 4 2 . 2 4 6 4 2 . .
. 2 4 6 8 10 8 6 4 2 . 2 4 6 8 6 4 2 .
. . 2 4 6 8 6 4 2 . 2 4 6 8 10 8 6 4 2
. . . 2 4 6 4 2 . 2 4 6 8 101210 8 6 4
. . . . 2 4 2 . 2 4 6 8 1012141210 8 6
. . . . . 2 . 2 4 6 8 10121416141210 8
. . . . . . 2 4 6 8 101214161816141210
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. . . . . . . . . 2 4 6 8 101210 8 6 4
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. . . . . . . . . . . . 2 4 6 4 2 . .
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. . . . . . . . . 2 4 6 4 2 . . . . .
2 . . . . . . . . . 2 4 2 . . . . . .
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2 . 2 . . . . . . . . . . . . . 2 4 2
. 2 4 2 . . . . . . . . . . . . . 2 .
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. . . . . . . . . 2 4 6 4 2 . . . . .
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. . . . . . . 2 4 2 . 2 . . . 2 . . .
. . . . . . 2 4 6 4 2 . . . . . . . .
. . . . . . . 2 4 2 . . . . . . . . .
. . . . . . . . 2 . . . . . . . . . 2
. . . . . . . . . . . . . . . . . . .
. . 2 . . . . . . . . . . . . . . . .
. 2 4 2 . . . . . . . . . . . . . . .
2 4 6 4 2 . . . . . . . . . . . . . 2
. 2 4 2 . . . . . . . . . . . . . 2 .
2 . 2 . . . . . . . . . . . . . 2 4 2
4 2 . . . . . . . . . . . . . 2 4 6 4
2 . . . . . . . . . . . . . . . 2 4 2
. . . . . . . . . . . . . . . . . 2 .
. . . . . . . . . . . . . . . . . . .
. 2 . . . . . . . . . 2 . . . . . . .
2 . . . . . . . . . 2 4 2 . . . . . .
. . . . . . . . . 2 4 6 4 2 . . . . .
. . . . 2 . . . 2 . 2 4 2 . . . . . .
. . . 2 . . . 2 4 2 . 2 . . . . . . .
. . . . . . 2 4 6 4 2 . . . . . . . .
. 2 . . . . . 2 4 2 . . . . . 2 . . .
2 . . . . . . . 2 . . . . . . . 2 . .
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80
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.
Let γ := T3|(2,0) α = T9|(5,0) δ. Let R(1) (x, y) := (1 − x, y) and R(2) (x, y) :=
(x, 1 − y), and for any function f : [0, 1]2 −→ R we denote f ◦ R(1) , f ◦ R(2) ,
and f ◦ R(1) ◦ R(2) by f 1 , f 2 and f 12 , respectively. From the above graph we
can conjecture the following blow-up rules for α:
92
γ
γ1
1 1
β
3
γ1
1
α
3
γ1
1 1
β
3
γ1
γ
7−→
α
Since β is a slice of the fractal associated to T3|0 [a] = [(x3 , y 3 )]—a regular
even class with parameter t = —in order to verify that T3|(0,0) α = T3|(2,2) α =
1 1
β
3
we calculate the image of the ideal class of b = (x3 , y 3 ) under R2 ◦ R1 .
As in the previous example, let H = xy(x + y)(x + y).
(x3 , y 3 ) ∼
=
=
x4
x3 (x − y), y 3 (x − y)
x3 (x − y) + H, y 3 (x − y)
+
2 x2 y 2
+
xy 3 , y 3 (x
− y) .
Now y 3 (x − y) ≡ 7 x2 y 2 + 2 xy 3 (mod `2 `3 `4 ), and therefore R1 ([b]) is the
3
2
2
3
7
2
2
2
3
class of the ideal x + xy + y , x y + xy . Then
x3 + 2 xy 2 + y 3 , 7 x2 y 2 + 2 xy 3
=
=
x3 + 2 xy 2 + y 3 , 7 x2 y 2 + 2 xy 3 + H
x3 + 2 xy 2 + y 3 , x3 y + 5 x2 y 2 + 3 xy 3 ,
and since x3 + 2 xy 2 + y 3 ≡ 6 x2 y + xy 2 + y 3 (mod `1 `3 `4 ), R2 ◦ R1 ([b]) is
the ideal class represented by 6 x2 y + xy 2 + y 3 , x3 + 5 x2 y + 3 xy 2 . Thus
R2 ◦ R1 ([b]) has invariant t = 7 .
A simple calculation now shows that T9|(3,3,0,0) [a] and T9|(5,5,2,2) [a] also
have invariant 7 , and therefore T3|(0,0) α = T3|(2,2) α = 31 β 1 .
93
The fractal γ = T3|(2,0) α is a two dimensional slice of the fractal attached
to T9|(5,5,0,0) [a] = [(x4 , y 4 )], a regular class with invariant t = 4 = −1. One
verifies that T9|(3,3,2,2) [a] is also regular and has invariant −1, so T3|(0,2) α is
also γ.
Now in order to verify that T3|(0,1) α = T3|(1,0) α = T3|(2,1) α = T3|(1,2) α = γ 1
we calculate the image of the ideal class represented by (x4 , y 4 ) under R2 ◦R1 .
A calculation similar to the one above shows that this reflection is represented
by the ideal
7 x2 y 2 + 2 xy 3 + x3 + 2 x2 y + xy 2 , 3 x3 y + 3 x2 y 2 + 2 xy 3 + y 4 ,
and therefore has invariant t = 2 . We conclude by verifying that 2 is also
the invariant of the transforms of [a] under T9|(3,3,1,1) , T9|(4,4,0,0) , T9|(5,5,1,1) and
T9|(4,4,2,2) .
Finally, to verify that T3|(1,1) α = 13 α, we simply check that the invariant
of T9|(4,4,1,1) [a] is 5 .
94
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. 2 4 6 4 2 . . . . . . . . . . . . . . . . . . . 2
2 4 6 8 6 4 2 . . . . . . . . . . . . . . . . . 2 .
2
2
2 4 6 8 10 8 6 4 2 . . . . . . . . . . . . . . . 2 4 2
2
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2
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.
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.
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2
.
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. 2 4 6 8 1012141618202224262830323436383634323028262422201816141210 8 6 4 2 . . . .
.
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.
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2 4 6 4 2
.
.
6 8 1012141618202224262830323436384042444240383634323028262422201816141210 8 6 4 2 .
2 . 2 4 2 .
2
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2
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.
.
6 4 2 . . .
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.
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6 8 1012141618202224262830323436384042444648504846444240383634323028262422201816141210 8 6
4 2 . . . .
.
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. . . . . .
.
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.
.
. . . . . . 2 . 2 4 6 8 10121416182022242628303234363840424446485052504846444240383634323028262422201816141210 8
. . . . . 2 4 2 . 2 4 6 8 1012141618202224262830323436384042444648504846444240383634323028262422201816141210 8 6
.
2
.
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.
.
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.
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.
.
. 2 4 6 8 101210 8 6 4 2 . 2 4 6 8 101214161820222426283032343638404240383634323028262422201816141210 8 6 4 2 . .
.
.
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2 4 6 8 1012141210 8 6 4 2 . 2 4 6 8 10121416182022242628303234363840383634323028262422201816141210 8 6 4 2 . . .
.
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4 6 8 10121416141210 8 6 4 2 . 2 4 6 8 1012141618202224262830323436383634323028262422201816141210 8 6 4 2 . . . .
.
.
6 8 101214161816141210 8 6 4 2 . 2 4 6 8 101214161820222426283032343634323028262422201816141210 8 6 4 2 . . . . .
.
2
8 1012141618201816141210 8 6 4 2 . 2 4 6 8 10121416182022242628303234323028262422201816141210 8 6 4 2 . . . 2 . .
.
. 2 4 6 8 10121416182022201816141210 8 6 4 2 . 2 4 6 8 1012141618202224262830323028262422201816141210 8 6 4 2 . . . 2 . . .
.
2 4 6 8 101214161820222422201816141210 8 6 4 2 . 2 4 6 8 101214161820222426283028262422201816141210 8 6 4 2 . . . . . . . .
.
2 4 6 8 1012141618202224262422201816141210 8 6 4 2 . 2 4 6 8 10121416182022242628262422201816141210 8 6 4 2 . . . . . . . . .
.
4 6 8 10121416182022242628262422201816141210 8 6 4 2 . 2 4 6 8 1012141618202224262422201816141210 8 6 4 2 . . . . . . . . . .
.
6 8 101214161820222426283028262422201816141210 8 6 4 2 . 2 4 6 8 101214161820222422201816141210 8 6 4 2 . . . . . . . . . . .
.
8 1012141618202224262830323028262422201816141210 8 6 4 2 . 2 4 6 8 10121416182022201816141210 8 6 4 2 . . . . . 2 . . . . . .
. 2 4 6 8 10121416182022242628303234323028262422201816141210 8 6 4 2 . 2 4 6 8 1012141618201816141210 8 6 4 2 . . . . . . . 2 . . . . .
2 4 6 8 101214161820222426283032343634323028262422201816141210 8 6 4 2 . 2 4 6 8 101214161816141210 8 6 4 2 . . . . . . . . . . . . . .
4 6 8 1012141618202224262830323436383634323028262422201816141210 8 6 4 2 . 2 4 6 8 10121416141210 8 6 4 2 . . . 2 . . . . . . . . . 2 .
6 8 10121416182022242628303234363840383634323028262422201816141210 8 6 4 2 . 2 4 6 8 1012141210 8 6 4 2 . . . 2 . . . . . . . . . 2 4 2
2 4 6 8 101214161820222426283032343638404240383634323028262422201816141210 8 6 4 2 . 2 4 6 8 101210 8 6 4 2 . . . . . . . . . . . . . 2 4 6 4
4 6 8 1012141618202224262830323436384042444240383634323028262422201816141210 8 6 4 2 . 2 4 6 8 10 8 6 4 2 . . . . . . . . . . . . . . . 2 4 2
6 8 10121416182022242628303234363840424446444240383634323028262422201816141210 8 6 4 2 . 2 4 6 8 6 4 2 . . . . . . . . . . . . . . . . . 2 .
8 101214161820222426283032343638404244464846444240383634323028262422201816141210 8 6 4 2 . 2 4 6 4 2 . . . . . . . . . . . . . . . . . . . 2
4 6 8 1012141618202224262830323436384042444648504846444240383634323028262422201816141210 8 6 4 2 . 2 4 2 . . . . . 2 . . . . . . . . . 2 . . . . .
6 8 10121416182022242628303234363840424446485052504846444240383634323028262422201816141210 8 6 4 2 . 2 . . . . . . . 2 . . . . . . . 2 . . . . . .
8 101214161820222426283032343638404244464850525452504846444240383634323028262422201816141210 8 6 4 2 . . . . . . . . . . . . . . . . . . . . . . .
6 8 10121416182022242628303234363840424446485052504846444240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2
4 6 8 1012141618202224262830323436384042444648504846444240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 4 2 . . . . . 2 . . . . . . . . .
2 4 6 8 101214161820222426283032343638404244464846444240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 4 6 4 2 . . . . . . . . . . . . . .
. 2 4 6 8 10121416182022242628303234363840424446444240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 . 2 4 2 . . . 2 . . . . . 2 . 2 . . .
. . 2 4 6 8 1012141618202224262830323436384042444240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 4 2 . 2 . . . 2 . . . . . 2 . . . 2 . .
. . . 2 4 6 8 101214161820222426283032343638404240383634323028262422201816141210 8 6 4 2 . . . . . . . 2 4 6 4 2 . . . . . . . . . . . . . . . . .
. . . . 2 4 6 8 10121416182022242628303234363840383634323028262422201816141210 8 6 4 2 . . . . . 2 . . . 2 4 2 . . . . . . . . . 2 . . . . . . . 2
. . . . . 2 4 6 8 1012141618202224262830323436383634323028262422201816141210 8 6 4 2 . . . . . . . 2 . . . 2 . . . . . . . . . 2 . . . . . . . . .
. . . . . . 2 4 6 8 101214161820222426283032343634323028262422201816141210 8 6 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 2 . . . 2 4 6 8 10121416182022242628303234323028262422201816141210 8 6 4 2 . . . 2 . . . . . . . . . . . 2 . . . . . 2 . . . . . . . . . . .
. . . . 2 . . . 2 4 6 8 1012141618202224262830323028262422201816141210 8 6 4 2 . . . 2 . . . . . . . . . . . 2 . . . . . 2 . . . . . . . . . . . 2
. . . . . . . . . 2 4 6 8 101214161820222426283028262422201816141210 8 6 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 2 4 6 8 10121416182022242628262422201816141210 8 6 4 2 . . . . . . . . . . . . . . . 2 . . . . . 2 . . . . . . . . . . . 2 . .
. . . . . . . . . . . 2 4 6 8 1012141618202224262422201816141210 8 6 4 2 . . . . . . . . . . . . . . . 2 . . . . . 2 . . . . . . . . . . . 2 . . .
. . . . . . . . . . . . 2 4 6 8 101214161820222422201816141210 8 6 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2 . . . . . 2 4 6 8 10121416182022201816141210 8 6 4 2 . . . . . 2 . . . . . . . . . 2 . . . . . 2 . . . . . . . . . . . 2 . . . . .
. . . . . . 2 . . . . . . . 2 4 6 8 1012141618201816141210 8 6 4 2 . . . . . . . 2 . . . . . . . 2 . . . . . 2 . . . . . . . . . . . 2 . . . . . 2
. . . . . . . . . . . . . . . 2 4 6 8 101214161816141210 8 6 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 2 . . . . . . . . . 2 . . . 2 4 6 8 10121416141210 8 6 4 2 . . . 2 . . . . . . . . . 2 . . . . . . . . . 2 . . . . . . . . . 2 . . . . . . . .
. 2 4 2 . . . . . . . . . 2 . . . 2 4 6 8 1012141210 8 6 4 2 . . . 2 . . . . . . . . . 2 4 2 . . . . . . . . . 2 . . . . . . . 2 . . . . . . . . .
2 4 6 4 2 . . . . . . . . . . . . . 2 4 6 8 101210 8 6 4 2 . . . . . . . . . . . . . 2 4 6 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . 2
. 2 4 2 . . . . . . . . . . . . . . . 2 4 6 8 10 8 6 4 2 . . . . . . . . . . . . . . . 2 4 2 . 2 . . . 2 . . . . . 2 . . . 2 . . . . . 2 . . . 2 .
2 . 2 . . . . . . . . . . . . . . . . . 2 4 6 8 6 4 2 . . . . . . . . . . . . . . . . . 2 . 2 4 2 . . . 2 . . . . . 2 . 2 . . . . . 2 . . . 2 4 2
4 2 . . . . . . . . . . . . . . . . . . . 2 4 6 4 2 . . . . . . . . . . . . . . . . . . . 2 4 6 4 2 . . . . . . . . . . . . . . . . . . . 2 4 6 4
2 . . . . . 2 . . . . . . . . . 2 . . . . . 2 4 2 . . . . . 2 . . . . . . . . . 2 . . . . . 2 4 2 . . . . . 2 . . . . . . . . . 2 . . . . . 2 4 2
. . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 . . . . . . . 2 .
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20
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80
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4 6 4 2
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2 . . .
4 2 . .
6 4 2 .
j
The above graph shows the function (i, j) 7−→ 243·β( 81i , 81
), and indicates
the following blow-up rules for β:
95
β
7−→
1 1
α
3
1 2
δ
9
1 12
δ
9
1 2
δ
9
X
1 1
δ
9
1
δ
9
1 1
δ
9
1 1
α
3
Here X seems to be a piecewise linear function, and that is in fact simple
to check: the fractal associated to T9|(1,1,1,1) [a] is the function x 7−→ max{2 −
d(x, 0), 2 − d(x, ε1234 )} (as seen in the previous example).
The fact that some transforms of β are equal to δ or reflections of δ can be
easily proved by looking at the corresponding transforms of [a], verifying that
they are quasi-regular, and looking at the corners at which the corresponding
fractals are equal to 2.
Finally, α is a slice of T3|(1,1,0,0) [a] = [(x2 , y 2 )], and to find out whether
T3|(2,0) β and T3|(0,2) β are in fact 31 α1 we could determine the reflection of the
class represented by the ideal (x2 , y 2 ) under R2 ◦ R1 . But since our base
field k in this case has 9 elements, there are 6 regular even ideal classes, and
hence, by exclusion, this reflection must have invariant t = 6 . Now one can
easily verify that 6 is also the invariant of both T9|(2,2,0,0) [a] and T9|(0,0,2,2) [a],
and this shows that T3|(2,0) β = T3|(0,2) β = 31 α1 .
j
The following graph shows the function (i, j) 7−→ 729 · γ( 81i , 81
):
96
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2 4
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4 6
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6 8
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8 10 8 6 4 2 . . .
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.
2 4 6 8 101210 8 6 4 2 . .
.
.
2 4 6 8 1012141210 8 6 4 2 .
.
4 6 8 10121416141210 8 6 4 2
2
.
.
6 8 101214161816141210 8 6 4
.
.
4 6 8 10121416141210 8 6 4 2
.
.
2 4 6 8 1012141210 8 6 4 2 .
.
2 4 6 4 2 . 2 4 6 8 101210 8 6 4 2 . .
.
4 6 8 6 4 2 . 2 4 6 8 10 8 6 4 2 . . .
. 2 4 6 8 10 8 6 4 2 . 2 4 6 8 6 4 2 . . . .
2 4 6 8 101210 8 6 4 2 . 2 4 6 4 2 . . . . .
4 6 8 1012141210 8 6 4 2 . 2 4 2 . . . . . 2
6 8 10121416141210 8 6 4 2 . 2 . . . . . . .
8 101214161816141210 8 6 4 2 . . . . . . . .
6 8 10121416141210 8 6 4 2 . . . 2 . . . . .
4 6 8 1012141210 8 6 4 2 . . . 2 . . . . . .
2 4 6 8 101210 8 6 4 2 . . . . . . . . . . .
. 2 4 6 8 10 8 6 4 2 . . . . . . . . . . . .
. . 2 4 6 8 6 4 2 . . . . . . . . . . . . .
. . . 2 4 6 4 2 . . . . . . . . . . . . . .
. . . . 2 4 2 . . . . . 2 . . . . . . . . .
. . . . . 2 . . . . . . . 2 . . . . . . . 2
. . . . . . . . . . . . . . . . . . . . . .
. . . 2 . . . . . . . 2 . . . . . . . 2 . .
. . . . 2 . . . . . 2 4 2 . . . . . 2 . . .
. . . . . . . . . 2 4 6 4 2 . . . . . . . .
. . . . . . . . 2 . 2 4 2 . . . 2 . . . . .
. . . . . . . 2 4 2 . 2 . . . 2 . . . . . 2
. . . . . . 2 4 6 4 2 . . . . . . . . . . .
. . . . . . . 2 4 2 . . . . . . . . . 2 . .
. . . . . . . . 2 . . . . . . . . . 2 . . .
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. . 2 . . . . . . . . . . . . . . . . . 2 .
. 2 4 2 . . . . . . . . . . . . . . . 2 4 2
2 4 6 4 2 . . . . . . . . . . . . . 2 4 6 4
. 2 4 2 . . . . . . . . . . . . . 2 . 2 4 2
2 . 2 . . . . . . . . . . . . . 2 4 2 . 2 .
4 2 . . . . . . . . . . . . . 2 4 6 4 2 . .
2 . . . . . . . . . . . . . . . 2 4 2 . . .
. . . . . . . . . . . . . . . . . 2 . . . .
. . . . . . . . . . . . . . . . . . . . . .
. 2 . . . . . . . . . 2 . . . . . . . . . 2
2 . . . . . . . . . 2 4 2 . . . . . . . . .
. . . . . . . . . 2 4 6 4 2 . . . . . . . .
. . . . 2 . . . 2 . 2 4 2 . . . . . . . . .
. . . 2 . . . 2 4 2 . 2 . . . . . . . . . .
. . . . . . 2 4 6 4 2 . . . . . . . . . . .
. 2 . . . . . 2 4 2 . . . . . 2 . . . . . .
2 . . . . . . . 2 . . . . . . . 2 . . . . .
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20
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2 4 2 .
2
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2 4 6 8 10 8 6 4 2
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6 8 1012141210 8 6
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. 2 4 6 8 10121416141210 8
.
2
2 4 6 8 1012141210 8 6
2 4 6 4 2 . 2 4 6 8 101210 8 6 4
2 4 6 8 6 4 2 . 2 4 6 8 10 8 6 4 2
2 4 6 8 10 8 6 4 2 . 2 4 6 8 6 4 2 .
2 4 6 8 101210 8 6 4 2 . 2 4 6 4 2 . .
4 6 8 1012141210 8 6 4 2 . 2 4 2 . . .
6 8 10121416141210 8 6 4 2 . 2 . . . .
8 101214161816141210 8 6 4 2 . . . . .
6 8 10121416141210 8 6 4 2 . . . 2 . .
4 6 8 1012141210 8 6 4 2 . . . 2 . . .
2 4 6 8 101210 8 6 4 2 . . . . . . . .
. 2 4 6 8 10 8 6 4 2 . . . . . . . . .
. . 2 4 6 8 6 4 2 . . . . . . . . . .
. . . 2 4 6 4 2 . . . . . . . . . . .
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. . . . . . . . . . . . 2 4 6 4 2 . .
. . . . . . . . . . . 2 4 6 8 6 4 2 .
. . . . . . . . . . 2 4 6 8 10 8 6 4 2
. . . . . . . . . 2 4 6 8 101210 8 6 4
. . . . 2 . . . 2 4 6 8 1012141210 8 6
. . . 2 . . . 2 4 6 8 10121416141210 8
. . . . . . 2 4 6 8 101214161816141210
. . . . . 2 . 2 4 6 8 10121416141210 8
. . . . 2 4 2 . 2 4 6 8 1012141210 8 6
. . . 2 4 6 4 2 . 2 4 6 8 101210 8 6 4
. . 2 4 6 8 6 4 2 . 2 4 6 8 10 8 6 4 2
. 2 4 6 8 10 8 6 4 2 . 2 4 6 8 6 4 2 .
2 4 6 8 101210 8 6 4 2 . 2 4 6 4 2 . .
4 6 8 1012141210 8 6 4 2 . 2 4 2 . . .
6 8 10121416141210 8 6 4 2 . 2 . . . .
8 101214161816141210 8 6 4 2 . . . . .
6 8 10121416141210 8 6 4 2 . . . 2 . .
4 6 8 1012141210 8 6 4 2 . . . 2 . . .
2 4 6 8 101210 8 6 4 2 . . . . . . . .
. 2 4 6 8 10 8 6 4 2 . . . . . . . . .
. . 2 4 6 8 6 4 2 . . . . . . . . . .
. . . 2 4 6 4 2 . . . . . . . . . . .
. . . . 2 4 2 . . . . . 2 . . . . . .
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40
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4 6 8 1012141210 8 6
. .
2 4 6 8 10121416141210 8
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4 6 8 101214161816141210
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2 4 6 8 10121416141210 8
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2 4 2 . 2 4 6 8 1012141210 8 6
.
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4 6 4 2
2 4 6 8 101210 8 6 4
. . 2 4 6 8 6 4 2 . 2 4 6 8 10 8 6 4 2
. 2 4 6 8 10 8 6 4 2 . 2 4 6 8 6 4 2 .
2 4 6 8 101210 8 6 4 2 . 2 4 6 4 2 . .
4 6 8 1012141210 8 6 4 2 . 2 4 2 . . .
6 8 10121416141210 8 6 4 2 . 2 . . . .
8 101214161816141210 8 6 4 2 . . . . .
6 8 10121416141210 8 6 4 2 . . . 2 . .
4 6 8 1012141210 8 6 4 2 . . . 2 . . .
2 4 6 8 101210 8 6 4 2 . . . . . . . .
. 2 4 6 8 10 8 6 4 2 . . . . . . . . .
. . 2 4 6 8 6 4 2 . . . . . . . . . .
. . . 2 4 6 4 2 . . . . . . . . . . .
. . . . 2 4 2 . . . . . 2 . . . . . .
. . . . . 2 . . . . . . . 2 . . . . .
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. 2 . . . . . . . 2 . . . . . . . 2 .
2 . . . . . . . . . 2 . . . . . 2 4 2
. . . . . . . . . . . . . . . 2 4 6 4
. . . . 2 . 2 . . . . . 2 . . . 2 4 2
. . . 2 . . . 2 . . . . . 2 . . . 2 .
. . . . . . . . . . . . . . . . . . 2
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2 . . . . . . . . . 2 . . . . . . . .
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2 . . . . . . . . . . . 2 . . . . . 2
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2 . . . . . 2 . . . . . . . . . . . 2
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2 . . . . . . . . . 2 . . . . . . . 2
4 2 . . . . . . . . . . . . . . . . .
2 . 2 . . . 2 . . . . . 2 . . . 2 . .
. 2 4 2 . . . 2 . . . . . 2 . 2 . . .
2 4 6 4 2 . . . . . . . . . . . . . .
. 2 4 2 . . . . . 2 . . . . . . . . .
. . 2 . . . . . . . 2 . . . . . . . 2
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60
4
2
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2 .
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2
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The above graph indicates the following blow-up rules:
γ
7−→
97
1
α
9
1
β
9
1
γ
3
1
β
9
Y
1
β
9
1
γ
3
1
β
9
1
α
9
2
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The occurrences of α, β and γ in the blow-up diagram can be verified by
looking at the invariants of the corresponding transforms of [a]. The function
Y = T3|(1,1) γ is identically zero—the fractal attached to the corresponding
class, T27|(16,16,1,1) [a], is 2 at ε13 and ε24 , and ≤ 1 at the other corners. So
it is piecewise linear, and Y turns out to be precisely the section where this
function vanishes.
Note the symmetry shown by α, β and γ across the SE-to-NW and SWto-NE diagonals, guaranteed by theorems 3.25 and 3.26.
98
Bibliography
[CH98]
L. Chiang and Y. C. Hung, On Hilbert-Kunz Functions of Some
Hypersurfaces, Journal of Algebra 199(1998), 499-527.
[HM93]
C. Han and P. Monsky, Some surprising Hilbert-Kunz functions,
Math. Zeitschrift 214(1993), 119-135.
[Mon83]
P. Monsky, The Hilbert-Kunz Function, Math. Annalen 263
(1983), 43-49.
99