UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLVII
2009
PENCILS OF PLANE CURVES AND THE NEWTON
POLYGON
by Andrzej Lenarcik and Mateusz Masternak
Anna Kowalska in memoriam
Abstract. Let f (X, Y ) ∈ C[X, Y ] be a polynomial of positive degree. Le
Van Thanh and Mutsuo Oka gave an estimation of the number of critical
values of the pencil f (X, Y ) − t (t ∈ C) at infinity in terms of the Newton
diagram of f . They used toric modifications. We reprove this result using
different methods. We propose more precise estimation close to the result
of Masaharu Ishikawa.
Introduction. Let us consider a polynomial of two variables f = f (X, Y ) ∈
C[X, Y ] of positive degree. This polynomial determines a pencil of projective
curves C t , t ∈ C, where C t is the projective closure of the affine curve f (X, Y )−
t = 0. It is well known that the set Λ∞ (f ) of t ∈ C for which the Milnor
numbers of curves C t at points from the line at infinity L∞ change is finite
(see for example [1]). We call Λ∞ (f ) the set of critical values of f at infinity.
Many authors in the last years made effort to decribe and to estimate this set
(see for example [2, 4, 5, 6, 7, 10, 11, 17, 18]). Ha Hui Vui and Pham Tien
Son [8] gave an exhaustive overwiew of a number of results in this subject.
In this article we focus attention on the results of Le Van Thanh and
Mutsuo Oka [14] and [15] who estimated the number of special values of Λ∞ (f )
in terms of the Newton diagram of f . The main role in this estimation is played
by the so called special faces (“bad” faces) of the Newton diagram which are
included in the lines going through the origin and not included in the axes.
2000 Mathematics Subject Classification. Primary 32S55; Secondary 14H20.
Key words and phrases. Plane curve, Newton polygon, pencil of plane curve singularities,
critical values, polar invariants.
The work is partially supported by the Polish MSHE grant No N N201 386634.
284
In [14] the numbers of factors different from X and Y of the initial forms
with respect to this faces are considered in the estimation. We propose an
improvement of this result by decreasing the number of factors that have to be
considered (Theorem 2.5). We obtain this result from stronger Theorem 2.1
(Main Result), where we compute special values associated to special faces
as critical values of some one-variable polynomials. This result is close to
the result of Masaharu Ishikawa [9]. In [15] the authors propose a stronger
estimation by replacing the number of factors of the initial forms by the number
of factors of the derivatives. This result can also be derived from Theorem 2.1.
Below, we present logical dependences between quoted and proposed theorems.
Le-Oka [15]
Le-Oka [14]
.
-
-
.
Theorem 2.5
Theorem 2.1
Le Van Thanh and Mutsuo Oka use toric modifications. We use a different
technique from [3, 13] and [16]. The Newton diagram of the curve f = 0
of degree N is included in the triangle with vertices (0, 0), (N, 0) and (0, N ).
When we treat (N, 0) as a new origin we can see in fact a local Newton diagram
of the projectivization of the curve at the point at infinity (1 : 0 : 0) ∈ P2 (C).
Analogously (0, N ) corresponds to (0 : 1 : 0) ∈ P2 (C). To prove Theorem 2.1
we use a local result (Theorem 3.1) to these two points at infinity.
There exist precise characterizations of Λ∞ (f ) in the two dimensional case
(see [8] for further references). For example, Masaharu Ishikawa in [9] completed the idea of Le Van Thanh and Mutsuo Oka and gave formulas for Λ∞ (f )
using Newton diagrams and toric modifications. E. R. Garcı́a Barroso and
A. Ploski [3] proposed a method of determining Λ∞ (f ) by localizing f (X, Y )
at points at infinity. Then they used factorization of the polar curve and the
so called polar invariants [19]. Since we know how to determine the polar invariants from the local Newton polygon [13], we apply their method to prove
Theorem 3.1.
The structure of the paper is as follows. In the next section we give a
definition of special values at infinity. Then we present in Section 2 our results
with comparison to the results of Le Van Thanh and Mutsuo Oka. Then we
prove a local result in Section 3. We finish with the proof of the Main Result
in Section 4.
285
1. Critical values at infinity. Let f (X, Y ) be as above and let N =
deg f > 0. We can write
X
(1)
f (X, Y ) =
cαβ X α Y β ,
cαβ ∈ C .
α+β≤N
Let us denote by f + (X, Y ) the leading form of f which is the sum of monomials
appearing in (1) of degree N . We consider the homogenization of f :
(2)
F (X, Y, Z) = c00 Z N + (c10 X + c01 Y )Z N −1 + · · · + f + (X, Y ) ,
that defines the projective curve
C = {(x : y : z) ∈ P2 (C)| F (x, y, z) = 0}
which is the projective closure of the affine curve f = 0. Let L∞ ⊂ P2 (C)
be the line at infinity (described by Z = 0). Since deg f > 0 the curve C has
finite set of points at infinity C∞ = C ∩ L∞ . From (2) we have
C∞ = {(x : y : 0) ∈ P2 (C) : f + (x, y) = 0} .
For every t ∈ C we consider the affine curve f (X, Y ) − t (which may have
multiple components) and its projective closure C t ⊂ P2 (C) given by the
equation F (X, Y, Z) − tZ N = 0. Clearly (C t )∞ = C∞ for every t ∈ C.
For t ∈ C and P ∈ C t we denote by µtP = µP (C t ) the Milnor number
of the curve C t at the point P . For every P ∈ C∞ there exits the number
for almost all
for all t ∈ C and µtP = µgen
≥ 0 such that µtP ≥ µgen
µgen
P
P
P
t > µgen } is finite as well as the set
t ∈ C. That
is
the
set
Λ
(f
)
=
{t
∈
C
:
µ
P
P
P
S
Λ∞ (f ) = P ∈C∞ ΛP (f ). This fact is due to Broughton [1] (see also [11] or [3]
for simple direct proofs).
2. Global results. The support supp f of the polynomial f (X, Y ) contains those (α, β) ∈ N2 that correspond to nonzero coefficients cαβ in (1). The
Newton diagram at infinity ∆∞ (f ) is the convex hull of {(0, 0)} ∪ supp f . We
also define ∆(f ) = convex(suppf ). Let us consider the set of lines that join
the origin (0, 0) with points of the support different from the origin. We denote by l0 (resp. l00 ) the only line from this set which has the minimal angle
with the horizontal axis (resp. vertical axis). The lines l0 and l00 can coincide
themeselves and with the axes. Let N∞ (f ) be the set of faces of ∆(f ) included
in the boundary of ∆∞ (f ) and not included in the axes. In this article the
term “face” means “1-dimensional face.” Let F = l0 ∩ ∆(f ) and L = l00 ∩ ∆(f );
F , L are faces or vertices of ∆(f ).
286
β
β
l00
L
∆(f )
l00
∆(f ) = ∆∞ (f )
l0
L
l0
F
F
α
f (0, 0) = 0
α
f (0, 0) 6= 0
Consider a set Z which is a face or a vertex of the boundary of ∆(f ). Let
r(Z) = (the number of lattice points on Z) − 1. We define the initial form
in(f, Z) of f as the sum of cαβ X α Y β appearing in (1) such that (α, β) ∈ Z.
For any quasi-homogeneous form ϕ ∈ C[X, Y ] we denote by r(ϕ) the number
of different irreducible factors of ϕ not equal to X and not equal to Y . We
put r(f, Z) = r (in(f, Z)). We have r(Z) ≥ r(f, Z). When Z is a vertex then
r(Z) = r(f, Z) = 0. If Z is a face and r(Z) = r(f, Z) then we say that f is
nondegenerate on Z. Let w be a polynomial of one variable. We call c ∈ C
a critical value of w when w − c has multiple factors. Due to this definition
a constant polynomial has no critical values. By crit w we denote the set of
critical values of w. Let wF (Y ) = in(f, F )(1, Y ) and wL (X) = in(f, L)(X, 1).
We use the convention that a sum over the empty set is zero (the symbol #
stands for the number of elements).
Theorem 2.1. (Main Result)
Let f be a nonconstant polynomial. Then
Λ∞ (f ) = S ∪ crit wF ∪ crit wL
X
where S ⊂ C and #S ≤
(r(S) − r(f, S)) .
S∈N∞ (f )
S6⊂l0 ,l00
Corollary 2.2 (cf. [9, Example 6.2] and [18]). Suppose that f is nondegenerate on every face S ∈ N∞ (f ) not included in l0 or l00 . Then
Λ∞ (f ) = crit wF ∪ crit wL .
As a consequence of Theorem 2.1 we can obtain an estimation of the number of critical values. We start with two simple properties.
Property 2.3. Let w(X) ∈ C[X] be a polynomial of positive degree. Let
s be the number of different roots of w(X). Then the number of nonzero
critical values of w is less than or equal to s − 1.
287
Property 2.4. Let u(X) ∈ C[X] be a polynomial of positive degree and
w(X) = u(X q ) where q is a positive integer. Then the nonzero critical values
of both polynomials are the same.
Now, let η(f ) = 1 when wF (Y ) or wL (X) has multiple factors and let
η(f ) = 0, otherwise. We have the following
Theorem 2.5. Suppose that f (0, 0) = 0 and ∆∞ (f ) has nonempty interior.
Then
X
X
#Λ∞ (f ) ≤
(r(S) − r(f, S)) +
r(f, S) + η(f ) ,
S6⊂l0 ,l00
S⊂l0 ,l00
where S runs over the faces from N∞ (f ). Moreover if η(f ) = 1 then 0 ∈
Λ∞ (f ).
Proof. Let us notice that 0 ∈ crit wF ∪ crit wL ⇔ η(f ) = 1. To finish the
proof it suffices to show for S ∈ N∞ (f ):
X
X
(i) #(crit wF \ {0}) ≤
r(f, S) and (ii) #(crit wL \ {0}) ≤
r(f, S).
S⊂l0
S⊂l00
We prove (i). Suppose that wF (Y ) has a nonzero critical value. Hence F is not
included in the horizontal axis. Because l0 6= l00 , F is also not included in the
vertical axis. Since f (0, 0) = 0 we can write wF (Y ) = Y b g(Y ), where b > 0.
We have deg g(Y ) > 0, therefore F is a face of N∞ (f ). We finish applying
Properties 2.3 and 2.4 to show that #(crit wF \ {0}) ≤ r(f, F ).
Theorem 3.4 of [14] can be derived from Theorem 2.5. We put ε(f ) = 1
when wF or wL has 0 as a multiple root or we put ε(f ) = 0, otherwise.
Corollary 2.6 ([14, Theorem 3.4, p. 414], [9, Corollary 6.6]). Suppose
that f (0, 0) = 0 and ∆∞ (f ) has a nonempty interior. Then
X
X
(3)
#Λ∞ (f ) ≤
(r(S) − r(f, S)) +
r(S) + ε(f ) ,
S6⊂l0 ,l00
S⊂l0 ,l00
where S runs over the faces from N∞ (f ).
Proof. It suffices to show
X
X
r(S) + ε(f ) .
r(f, S) + η(f ) ≤
S⊂l0 ,l00
S⊂l0 ,l00
Since r(f, S) ≤ r(S) we have only to consider the case when η(f ) = 1 and
ε(f ) = 0. But this means that wF or wL has a nonzero multiple root. Suppose
that this is wF . As in the proof of Theorem 2.5 we conclude that F is a face
of N∞ (f ) included in l0 and in(f, F ) has multiple factors different from X or
Y . That is r(f, F ) < r(F ).
288
Example 2.7. Let f (X, Y ) = X 2 Y 4 (XY 2 − 2)2 − X 4 Y 2 (X 2 Y − 2)2 .
β
L
S
∆(f )
F
α
We have N∞ (f ) = {F, S, L}, r(F ) = r(L) = 2, r(S) = 4, in(f, F ) =
−X 4 Y 2 (X 2 Y − 2)2 , in(f, S) = X 4 Y 4 (Y − X)(Y + X)(Y − iX)(Y + iX),
in(f, L) = X 2 Y 4 (XY 2 − 2)2 , r(f, F ) = r(f, L) = 1, r(S) = r(f, S) = 4,
ε(f ) = η(f ) = 1. By Corollary 2.6 #Λ∞ (f ) ≤ 5, by Theorem 2.5 #Λ∞ (f ) ≤ 3.
We have here crit wF = {0, −1} and crit wL = {0, 1}. From Theorem 2.1
Λ∞ (f ) = {−1, 0, 1}.
Remark 2.8. Le Van Thanh and Mutsuo Oka in [15, Theorem 4.17, p. 593]
proposed a stronger estimation by replacing r(S) in the second component of
(3) by r(∂in(f, S)/∂Y ) for S ⊂ l0 and by r(∂in(f, S)/∂X) for S ⊂ l00 . Clearly
this result can also be derived from Theorem 2.1.
3. Local result. In this section we present some local results that will be
useful in the proof of Theorem 2.1 in Section 4.
Let
Y } be a nonzero power series without constant term. Write
P f ∈ C{X,
f =
cαβ X α Y β ∈ C{X, Y } and suppf = {(α, β) ∈ N2 : cαβ 6= 0 }. The
Newton polygon N0 (f ) is the set of the compact faces of the boundary of the
convex hull ∆0 (f ) of the set suppf + N2 . We call ∆0 (f ) the local Newton
diagram of f . By δ(f ) we denote the distance between ∆0 (f ) and the horizontal axis. For every S ∈ N0 (f ) we denote by |S|1 and |S|2 the lenghts of
the projection of S on the horizontal and vertical axes, respectively. We call
|S|1 /|S|2 the inclination of the face S. Let α/α(S) + β/β(S) = 1 be the equation of the line containing S. Clearly α(S), β(S) are rational numbers and
α(S)/β(S) = |S|1 /|S|2 . If Z is a face or a vertex of ∆0 (f ) then we define r(Z),
in(f, Z), r(f, Z) as in the global case (see Section 2).
Now, suppose that f is not divisible by X and let us fix N > 0. Let
ft = f −tX N be a pencil with Λ0 (f ) as the set of special values (see Section 1).
We consider the set of lines going through (N, 0) and (α, β) ∈ suppf with
α < N . We choose the only one line l from this set that has the minimal angle
289
with the horizontal axis. Let T = l ∩ ∆0 (f ) ∩ {α ≤ N }, wT (Y ) = in(f, T )(1, Y )
(see picture after (5)).
Theorem 3.1. Let f ∈ C{X, Y } be a series such that p = (f, X)0 is a
positive integer. Then
Λ0 (f ) = S ∪ crit wT ,
X
where S ⊂ C and #S ≤
(r(S) − r(f, S)) .
S∈N0 (f )
α(S)<N
Before giving the proof let us recall an important lemma. This is a local
version of the description of critical values at infinity given in [14, pp. 410–
411]. Let f /h be a meromorphic fraction with coprime f, h ∈ C{X, Y } and
let g = g(X, Y ) ∈ C{X, Y } be irreducible power series such that g does not
divide h. Let (x(u), y(u)) ∈ C{u}2 , (x(0), y(0)) = (0, 0) be a parametrization
of the branch g = 0. Then we put
f (x(u), y(u)) f
(g) =
∈ C ∪ {∞}.
h
h(x(u), y(u)) u=0
By (f, g)0 = dimC C{X, Y }/(f, g) we denote intersection multiplicity of f = 0
and g = 0.
Lemma 3.2 ([17, Théorème 1], [3, Proposition 2.2]). Let f ∈ C{X, Y } be
a nonzero series without constant term such that {X = 0} 6⊂ {f = 0} and let
ft = f − tX N (t ∈ C) be a pencil. Then
f
(g) : g ∈ B ,
Λ0 (f ) =
XN
where B is the set of branches g = 0 of the polar curve ∂f /∂Y = 0 such that
(f, g)0 /(X, g)0 ≥ N .
In the proof of Theorem 3.1 we use a version of the factorization theorem
of the polar curve from [13, Theorem 1.1] or [12, Theorem 2.1 with ϕ = 0].
The differences are rather technical therefore we present here a sketch of the
proof referring the reader to [13] or [12] for details.
Lemma 3.3. (Factorization of the polar curve ∂f /∂Y )
Let f (X, Y ) ∈ C{X, Y } be a nonzero reduced series without constant term.
Suppose that {X = 0} 6⊂ {f = 0}. Then there exists a factorization
Y
∂f
=C
AS BS in C{X, Y }
∂Y
S∈N0 (f )
such that
290
(a) If δ = δ(f ) > 1 then C = Y δ−1 and C = 1, otherwise.
(b) If g is an irreducible factor of AS BS then
(4)
(f, g)0
≥ α(S)
(X, g)0
with the equality if and only if g divides AS .
(c) Moreover,
· r(BS ) ≤ r(S) − r(f, S),
|S|2
· (BS , X)0 =
(r(S) − r(f, S)),
r(S)
|S|2
r(f, S) + σ(S) ,
· (AS , X)0 =
r(S)
σ(S) = −1 when S touches the horizontal axis and σ(S) = 0, otherwise.
S
Let C{X}∗= n≥1 C{X 1/n } be the ring of Puiseux series. Let p = ord f (0, Y ) >
1. Let us consider the sets of Puiseux solutions Zer f = {y1 (X), . . . , yp (X)}
and Zer(∂f /∂Y ) = {z1 (X), . . . , zp−1 (X)} in C∗ {X} of the equations f = 0
and ∂f /∂y = 0, respectively. The orders of these solutions are described by
the Newton diagrams ∆0 (f ) and ∆0 (∂f /∂Y ). We define a solution z(X) ∈
Zer(∂f /∂Y ) to be of the first kind if ord z(X) appears as an order of solution
from Zer f and of the second kind , otherwise. Since the solutions of a cycle
are of the same kind we can consider the branches of the first kind and of the
second kind . Let S ∈ N0 (f ). We put wS (Y ) = in(f, S)(1, Y ). We need the
following simple proposition:
Proposition 3.4. (description of Puiseux solutions of ∂f /∂Y = 0)
Let z(X) ∈ Zer(∂f /∂Y ). Then
(I) If z(X) is of the first kind then:
(a) if ord z(X) = +∞ then δ(f ) > 1,
(b) if ord z(X) < +∞ then there exists S ∈ N0 (f ) such that z(X) =
aX |S|1 /|S|2 + . . . (a 6= 0) and wS0 (a) = 0.
(II) Solutions of the second kind exist if and only if both of the following
condition holds:
− the lowest face S = L touches the horizontal axis (ie. wL (0) 6= 0),
− ord(wL (Y ) − wL (0)) ≥ 2.
(III) The order of every second kind solution is strictly greater than all the
orders of solutions from Zer f .
Proof of Lemma 3.3. If g = Y is a first kind branch then δ = δ(f ) > 1
and we put C = Y δ−1 . Every first-kind-branch g different from Y is naturally
associated with a face S ∈ N0 (f ). For a corresponding Puiseux solution z(X) ∈
291
Zer(∂f /∂Y ) we have
z(X) = aX |S|1 /|S|2 + . . . ,
a 6= 0,
wS0 (a) = 0,
f (X, z(X)) = wS (a)X α(S) + . . .
Then we put this branch to AS if wS (a) 6= 0 and to BS if wS (a) = 0. Since
(f, g)0
= ord f (X, z(X)) ,
(X, g)0
the inequality (4) is satisfied.
If a second-kind-branch g exists then the lowest face L ∈ N0 (f ) touches
the horizontal axis, that is wL (0) 6= 0. We put such a branch to AL . For
a corresponding Puiseux solution z(X) ∈ Zer(∂f /∂Y ) we have ord z(X) >
|L|1 /|L|2 and therefore f (X, z(X)) = wL (0)X α(L) + . . . Consequently
(f, g)0
= ord f (X, z(X)) = α(L) .
(X, g)0
To prove (c) let us observe that when differentiating in(f, S) with respect
to Y , the power of every factor different from X decreases by one (see [13,
Lemma 4.1] for details).
Proof of Theorem 3.1. By Lemma 3.3 we can write ∂f /∂Y = G1 G2 G3 ,
where
Y
Y
Y
G1 =
BS ,
G2 = C
AS BS ,
G3 =
AS
S∈N0 (f )
S∈N0 (f )
S∈N0 (f )
α(S)<N
α(S)≥N
α(S)<N
(we use convention that product over empty set equals 1). Let Bi = branches
of Gi (i = 1, 2, 3). Let B be defined as in Lemma 3.2. Clearly B2 ⊂ B ⊂
N
B
P1 ∪ B2 . Let us denote N0 (f ) = {S ∈ N0 (f ) : α(S) < N }. Since #B1 ≤
S∈N N (f ) (r(S) − r(f, S)) to finish the proof it suffices to show that
0
(5)
{(f /X N )(g) : g ∈ B2 ) = crit wT ,
where T = l ∩ ∆0 (f ) ∩ {α ≤ N } as earlier.
l
l
∆0 (f )
∆0 (f )
T
l
T (N, 0)
∆0 (f )
T
(N, 0)
(N, 0)
292
When l coincides with the horizontal axis then we have empty sets on both
sides of (5). Let us consider other cases. To prove the right inclusion in (5) let
us choose
f
f (X, z(X)) c=
(g)
=
,
XN
XN
X=0
where z(X) ∈ Zer(∂f /∂Y ) is a corresponding Puiseux solution and g is a
corresponding branch. Suppose that c = 0. Then ord f (X, z(X)) > N . By
Lemma 3.3 at least one of the following condition holds:
• T is a face of N0 (f ) and ord BT > 0,
• Y 2 divides wT (Y ).
In both cases wT (Y ) has a multiple root, hence 0 ∈ crit wT . Now, suppose
that c 6= 0. We have f (X, z(X)) = cX N + . . . . If g is of the first kind then
g 6= Y . Then ord z(X) = |S|1 /|S|2 for some S ∈ N0 (f ) and S = T is the only
one possibility. We have z(X) = aX |T |1 /|T |2 + . . . and wT0 (a) = 0. Therefore
c = wT (a) ∈ crit wT . If g is of the second kind then T = L is the lowest
face of N0 (f ); α(T ) = N . Then wT (0) = 0 and ord(wT − wT (0)) ≥ 2. That
is wT0 (0) = 0. Since f (X, z(X)) = wT (0)X α(T ) + . . . in this case, we obtain
c = wT (0) ∈ crit wT .
In order to prove the left inclusion in (5) let us choose c ∈ crit wT . We
want to find g ∈ B2 such that (f /X N )(g) = c. Let us consider four cases.
c = 0, a = 0. In this case Y 2 divides wT (Y ) and by Lemma 3.3 Y is a
factor of ∂f /∂Y or there exist S ∈ N0 (f ) such that α(S) > N . Anyway, we
have z(X) ∈ Zer(∂f /∂Y ) such that f (X, z(X)) > N and we take as g the
corresponding branch.
c = 0, a 6= 0. Now, a is a nonzero multiple root of wT . Therefore T ∈ N0 (f )
and ord BT > 0. We choose as g a factor of BT .
c 6= 0, a = 0. In this case T touches the horizontal axis, wT (0) 6= 0, α(T ) = N .
Since wT0 (0) = 0 we have ord(wT − wT (0)) ≥ 0. Hence there exists a secondkind-solution g with corresponding z(X) ∈ Zer(∂f /∂Y ) such that
f
f (X, z(X)) (g) =
= wT (0) = c .
XN
XN
X=0
c 6= 0, a 6= 0. Now, wT (Y ) must have at least two different roots, that is
T ∈ N0 (f ). Because wT0 has nonzero root, there exists a face T̃ ∈ N0 (∂f /∂Y )
parallel to T such that
∂
in(f, T ) = in(∂f /∂Y, T̃ ) .
∂Y
293
By the Newton–Puiseux Theorem we choose a solution of ∂f /∂Y = 0 of the
form z(X) = aX |T |1 /|T |2 + . . . and take the corresponding branch g (it is a
factor of AT ).
4. Proof of Main Result. We use notation from Sections 2 and 3. Let
f = f (X, Y ) be a nonconstant polynomial as earlier (N = deg f > 0). We can
write
s
Y
(6)
f + (X, Y ) = (a monomial) (Y − ai X)νi ,
i=1
where ai are nonzero and pairwise different. We put
d(f ) =
s
X
i=1
(νi − 1)
and call d(f ) the degeneracy of f . If s = 0 then f + (X, Y ) reduces to a
monomial and d(f ) = 0.
Let us define Pa = (1 : a : 0) for a ∈ C and P∞ = (0 : 1 : 0). Then
L∞ = {Pa }a∈C∪{∞} . Let denote by ϕPa (U, V ) a localization of f to Pa defined
by ϕPa (U, V ) = F (1, a + V, U ) for a ∈ C and ϕP∞ (U, V ) = F (V, 1, U ). We
have
Pa ∈ C∞ ⇐⇒ ordV ϕPa (0, V ) = (C, L∞ )Pa > 0 .
Then (C, L∞ )Pai = νi for i = 1, . . . , s and (C, L∞ )P0 (resp. (C, L∞ )P∞ ) is
equal to the maximal power of Y (resp. of X) in (6). Our aim is to obtain a
partial description of the set
[
ΛP (f ),
Λ∞ (f ) =
P ∈C∞
where ΛP (f ) = Λ0 (ϕP (U, V )). We can write
[
Λ∞ (f ) =
ΛP (f ) ∪ ΛP0 (f ) ∪ ΛP∞ (f ) .
P ∈C∞
P 6=P0 ,P∞
Let us denote by S + the first component of the above union. Then we have
#S + ≤
s
X
i=1
(ordV ϕPai (0, V ) − 1) = d(f ) .
To describe the sets ΛP0 (f ) = Λ0 (ϕP0 (U, V )) = Λ0 (F (1, V, U )) and ΛP∞ (f ) =
Λ0 (ϕP∞ (U, V )) = Λ0 (F (V, 1, U )) we can use Theorem 3.1.
294
β
l00
N 00
W
L
∆(f )
N0
l0
F
α
Let W = convex(suppf + ). We define N + = {W } when W is a face and
N + = ∅ when W is a vertex. We have
N∞ (f ) \ {faces in l0 or l00 } = N 0 ∪ N + ∪ N 00 ,
where N 0 joins the right vertex of F with the right vertex of W and N 00 joins
the upper vertex of L with the upper vertex of W . Applying methods from
[16] (Lemma 2.1 and Corollary 2.2, p. 183) we can verify that there exists
one-to-one correspondences
and
N0N (ϕP0 ) = {S ∈ N0 (ϕP0 ) : α(S) < N } ←→ N 0
N0N (ϕP∞ ) = {S ∈ N0 (ϕP∞ ) : α(S) < N } ←→ N 00
that preserves the number of integral poins in the corresponding faces and the
number of factors different from variables in the corresponding initial forms.
Let denote by T 0 (resp. T 00 ) the intersection of the diagram ∆0 (F (1, V, U ))
(resp. ∆0 (F (V, 1, U ))) and the line supporting this diagram passing throught
(N, 0); T 0 , T 00 can be vertices or faces. Suppose that wT 0 and wT 00 are defined
as in Theorem 3.1. Then
ΛP0 (f ) = Λ0 (F (1, V, U )) = S 0 ∪ crit wT 0
295
and
ΛP∞ (f ) = Λ0 (F (V, 1, U )) = S 00 ∪ crit wT 00 ,
where
#S 0 ≤
X
(r(S) − r(ϕP0 , S)) =
S∈N N (ϕP0 )
X
S∈N 0
(r(S) − r(f, S))
and
#S 00 ≤
X
(r(S) − r(ϕP∞ , S)) =
S∈N N (ϕP∞ )
X
S∈N 00
(r(S) − r(f, S)).
Let us notice that when W is a face then r(W ) − r(f, W ) = d(f ). Therefore
for S = S 0 ∪ S + ∪ S 00 we obtain
X
#S ≤
(r(S) − r(f, S)).
S∈N∞ (f )
S6⊂l0 ,l00
Since wT 0 = wF and wT 00 = wL we obtain
which ends the proof.
Λ∞ (f ) = S ∪ crit wF ∪ crit wL
Acknowledgements. The authors wish to express their gratitude to Professor Arkadiusz Ploski for his inspiration as well as for his help during preparing this article.
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Received
December 12, 2009
Department of Mathematics
Technical University
Al. 1000 L PP 7
25-314 Kielce, Poland
e-mail : [email protected]
e-mail : [email protected]