ADDITIVE NUMBER THEORY SHEDS EXTRA
LIGHT ON THE HOPF-STIEFEL o FUNCTION
Author(en): Plagne, Alain
Objekttyp:
Article
Zeitschrift:
L'Enseignement Mathématique
Band(Jahr): 49(2003)
Heft 1-2:
L'ENSEIGNEMENT MATHÉMATIQUE
Erstellt am: 10 déc. 2012
Persistenter Link: http://dx.doi.org/10.5169/seals-66682
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ADDITIVE NUMBER THEORY SHEDS
EXTRA LIGHT ON THE HOPF-STIEFEL o FUNCTION
by
ABSTRACT.
famous
The
Alain Plagne
Hopf-Stiefel
function appears
o
in
several
places
in
)• However,
mathematics (linear and bilinear algebra, topology, intercalate matrices,
although the object of much study, this function kept a part of mystery since no simple
formula was known for it. We shall dérive a simple and practical explicit formula
.
.
.
generalized function due to
combinatorial group theory
proof relies on a
which follows from additive number theoretical arguments. It is shown that this last
resuit generalizes earlier ones by Eliahou and Kervaire and by Yuzvinsky.
for
more generally for
and
o
(3
P
arbitrary prime),
(p
Eliahou and Kervaire. The
new resuit in
Introduction
1.
composition formula of size [r,
of characteristic différent from
A
to be
where
(y\
,
.
.
Z\,z 2
.
,
y
s
)
,
,...,z
n
are
a
n
s,
2)
n]
is
over some field F (that we assume
an identity of the form
bilinear forms in the variables
with coefficients in
F
.
(jci
.
.
,
Jt r )
and
For example, the law of moduli for complex
numbers provides the identity
which
composition formula of size [2,2,2]. The law of moduli for
quaternions (respectively for octonions) provides a composition formula of
size [4,4,4] (respectively [8,8,8]) in a similar
way. Conversely, Hurwitz's
is
a
theorem [7] (see also [8]) states that the only possible values of
a composition formula of size
[n,n,ri\ exists are 1,2,4 and 8.
n
for which
Hurwitz problem, Hopf [6] and Stiefel
[15] began to study real division algebras. They introduced a function of an
algebraic nature. The so-called Hopf-Stiefel function o (which dépends on
two positive intégral variables r and s) is defined by the formula
In the late thirties, starting from the
(1.1)
Stiefel obtained
resuit which says that if a nonsingular bilinear
r
map from R xRs onto R" exists over R, then n > ros (Hopf-Stiefel
theorem). The link between nonsingular bilinear maps and o is realized by
Hopf
and
a
passing to projective spaces and their cohomology rings.
More generally, the o function appears in différent parts of mathematics.
It allows a nice mathematical trip: starting from bilinear algebra (Hurwitz 's
problem, Pfister's quadratic forms [10, 11]) and algebra (its basic définition
(1.1)), we pass through topology (Hopf-Stiefel theorem, real division algebras
and Yuzvinsky's theorem [17]), visit the theory of intercalate matrices
and
(Yuzvinsky's conjecture [17, 16])
arrive
at
additive number theory ([s]
The reader interested in the many applications of o
is mainly referred to the nice survey [13] by Shapiro, to his récent book [14,
Chapters 12-15] and to the book by Rajwade [12], especially Chapter 13.
and the présent paper).
With the algebraic viewpoint (1.1), it is fairly natural to generalize, with
Eliahou and Kervaire [s], the o function in the following way. For any given
prime p, we set
Evidently, /3 =
extension of (1.1)
2
o.
is
It
is
a
theorem of Eliahou and Kervaire
[5]
that
this
still relevant in the additive number theoretical context.
more generally of (3 P were known
(for instance recursion formulas or expression in terms of p-adic expansions),
describing thèse fonctions efficiently appeared difficult, so that it»is not rare
Although many properties of
to
see
o
and
tables giving the flrst values of o.
In this paper, perhaps surprisingly, we shall dérive a simple and practical
explicit formula for o and more generally for (3 p (p arbitrary prime).
THEOREM
the
1
.
Let
pbe
any prime number.
The
function
(3
is
P
given by
formula
Notice that clearly, the minimum involved
satisfying 0 < t < [log(max(r, s))/ logp]
.
is
attained for
a
value
of
t
In
particular, this resuit élucidâtes, in some sensé, the Hopf-Stiefel function.
THEOREM
2.
The
Hopf-Stiefel function
is
With this formula, a large number of properties of o follow immediately
or admit a simplified proof.
In fact Theorem 1 follows from an additive number theoretical theorem
(Theorem 3 below) which is of independent interest. Given a finite Abelian
group G, we derme (as in [s]) the function of two intégral variables r and s
(l<r,j<|G|)
where A + B
is
the usual
Minkowski sum of
sets, namely
arbitrary Abelian group G, determining the function jic is not easy:
this is an open problem in gênerai for G finite and Abelian. However, Eliahou
and Kervaire [5] obtained such a resuit for finite groups of prime exponent,
thus generalizing Yuzvinsky's resuit [17] for binary spaces.
Using a différent approach (based on Kneser's theorem [9]), we shall obtain
a resuit valid for any cyclic group.
Given
an
THEOREM
3.
Taking
= p,
n
Let
n
a
be
prime,
Moreover, as will
of Eliahou and Kervaire.
[2, 3, 4].
any integer.
be
If 1< r,s <n,
we hâve
exactly the Cauchy-Davenport theorem
explained in Section 3, this resuit contains that
gives
Since cyclic groups are characterized by the equality expG = |G|, this
theorem is a direct conséquence of the following more gênerai resuit.
THEOREM
we hâve
4.
Let G be any finite Abelian group. Then, if
1
<
r,s
<\G\,
With thèse results, we know the behaviour of
ijl g
the
at
two endpoints
of the
spectrum (cyclic groups and groups of prime exponent). What now
remains to be done is to fill the gap between the upper bound and the lower
bound for gênerai finite Abelian groups.
2.
Proof of Theorem
4
Let G be any given finite Abelian group and let
<
r,s < \G\.
lower bound
The
2.1
1
If HG(r,
s) >
r+s-1,
immédiate (take
the resuit is
d=
1). We may thus
assume that
(2.1)
Then, choosing two sets A and B in G with respective cardinalities
s, such that |^4 + £>| attains /j, G (r,s), we get
We are in
with
a
a
position
to
r
and
apply Kneser's theorem [9] on the structure of sets
small sumset. It follows that there exists
a
subgroup H of
G
(namely
the stabilizer of A + B) such that
Denoting by (A + H)/H (resp. (B + H)/H) the /7-cosets that A (resp. B)
intersects, we obtain
where
/
/
divides
dénotes the cardinality of H
\G\, we get
From this it follows that, in any case,
which
is
the desired lower bound.
.
Since Lagrange's theorem shows that
The upper bound
2.2
any subgroup of G. Choose Ao and
[>/|#|] and [s/ I^ll and sucn mat
Hbe
Let
cardinalities
of cardinality
Now choose
and
r
£>
BQB
Q
in G/ H with respective
of cardinality
smG
image of A (resp. B) by the canonical projection on G/H
(resp.
£>o).
is
such that the
included in Ao
One has
This proves the first lemma we need.
LEMMA
1
.
For any finite Abelian group
G
The second useful point is synthesized in the next
finite Abelian group. For any positive integer
following two propositions are équivalent
LEMMA
the
folkloric lemma.
(i)
(ii)
m
2.
Let G be
a
m,
divides expG,
there exists
subgroup
a
HofG
such that G/ H is isomorphic to
X/niL.
cyclic group X, trivial considérations (take two sets with
consécutive éléments), show that, for any u,v < \K\,
In the case of
a
(2.2)
Using consecutively Lemma
following chain of inequalities
inequality (2.2) and Lemma
1,
2
yields the
:
The change of variable
restrictions
J|L |d
an d
d —
j|
\G\/f yields
\q\
;
m
is
a
parameter
d
subject to the two
proves the upper bound in Theorem
4.
From Theorem 3 to Theorem
3.
1
We first use the theorem of Eliahou and Kervaire (see Section
which states that if
is
p
arbitrary prime,
an
r
and
s
3
of [s]),
two integers, then
(3.1)
whenever
d
p
> r,s
Now, from Theorem
soon
G
and
G
as
of [I], it follows that \iq coincides with \iqi as
are two Abelian p-groups of the same order. In other
10
words,
(3.2)
We would like to emphasize that from our method (more precisely, using
simply Lemma
together with
inductive argument (the quotient groups of
d
(Z/pZ) hâve the same form), we are able to dérive a simple direct (that is,
without using [1]) alternative proof of (3.2). Indeed, the only thing to verify
is
1)
an
that if
(3.3)
k>
for any
1
then we can construct sets A and B of respective cardinalities
=
r+
s
r
(resp.
the
s)
of the lexicographie
order.
and
r
(resp.
\
t
A +
s
with
for
B)
B\=r
+
|^4 +
the
£>|
—
l. This is achieved by taking for A
smallest possible éléments in the sensé
Hypothesis (3.3) then ensures that, in this case,
s-l.
We are now ready to prove Theorem
(but any
sufficiently large
Theorem
3,
we obtain
which proves Theorem
3.
d
1
.
We put for
instance
d —
will do). Using consecutively (3.1), (3.2)
r+s
and
function by Shalom
Eliahou during the conférence "Multilinear Algebra and Matroid Theory" held
in Lisboa, Portugal (March 24-26, 2002). There, he asked me for a formula
a question which led me to Theorem 3. He then kindly indicated
for
Z
ACKNOWLEDGEMENTS.
/i
introduced
was
I
to
the
\i
/nZ,
fonctions. The author is grateful to Shalom
Eliahou for his help as well as for a carefol reading of preliminary versions
of this paper. I would like to associate Michel Kervaire to thèse thanks, for
to
me the
application
the
to
(5
P
the interest he took in this paper.
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YUZVINSKY,
S.
(Reçu
Alain Plagne
LIX
École polytechnique
F-91128 Palaiseau Cedex
France
e-mail : [email protected]
le
6
J.
décembre 2002)