Chevalley’s theorem with restricted variables

Abstract

First, a generalization of Chevalley's classical theorem from 1936 on polynomial equations f(x1,…, xN) = 0 over a finite field K is given, where the variables xi are restricted to arbitrary subsets Ai ⊆ K. The proof uses Alon's Nullstellensatz. Next, a theorem on integer polynomial congruences f(x1,…, xN) ≡ 0 (mod pν) with restricted variables is proved, which generalizes a more recent result of Schanuel. Finally, an extension of Olson's theorem on zero-sum sequences in finite Abelian p-groups is derived as a corollary.

Full text

Chevalley’s theorem with restricted variables
David Brink
10 May 2010
First, a generalization of Chevalley’s classical theorem from 1936 on
polynomial equations f(x1, . . . , xN) = 0 over a finite field Kis given,
where the variables xiare restricted to arbitrary subsets Ai⊆K.
The proof uses Alon’s Nullstellensatz. Next, a theorem on integer
polynomial congruences f(x1, . . . , xN)≡0 (mod pν) with restricted
variables is proved, which generalizes a more recent result of Schanuel.
Finally, an extension of Olson’s theorem on zero-sum sequences in finite
Abelian p-groups is derived as a corollary.
Chevalley’s theorem states that a set of polynomials fj(X1,...XN) over a
finite field Kwithout constant terms has a non-trivial common zero
(a1, . . . , aN)∈KNif the number of variables Nexceeds the sum of
total degrees Pjdeg(fj), thereby settling in the affirmative the conjecture
of Artin that finite fields are quasi-algebraically closed [2].
Alon’s Nullstellensatz is the assertion that a polynomial f(X1, . . . , XN)
over an arbitrary field Kcannot vanish on a set QN
i=1 Aiwith Ai⊆Kif
it has a non-zero term δXt1
1· · · XtN
Nof maximal total degree and such that
|Ai|> tifor all i[1, Theorem 1.2]. This simple principle has become an
all-conquering force in combinatorics with applications in numerous areas.
Interestingly, the special case where Kis finite and Ai=Kfor all iappears
already in Chevalley’s proof.
The first theorem in the present note extends Chevalley’s theorem to
polynomials with restricted variables. The proof follows that of Chevalley,
but uses Alon’s Nullstellensatz rather than the above-mentioned special case
(see also [1, Theorem 3.1]).
Theorem 1. Consider the polynomials f1(X1, . . . , XN), . . . , fR(X1, . . . , XN)
over the finite field Fq. Let A1, . . . , ANbe non-empty subsets of Fqsuch that
X
i
(|Ai| − 1) >X
j
deg(fj)(q−1).
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Then the solution set
V={a∈Y
i
Ai|fj(a) = 0 for all j}
with the variables restricted to the Aiis not a singleton.
Proof. Assume for a contradiction V={a}with a= (a1, . . . , aN). Then
P(X) = Qj(1 −fj(X)q−1) satisfies
P(x) = (1 for x=a,
0 for x∈QiAi\{a}.
Further, Q(X) = QiQb∈Ai\{ai}(Xi−b) satisfies
Q(x) = (δfor x=a,
0 for x∈QiAi\{a},
with some (non-zero) δ∈Fq. It follows from the theorem’s assumption that
X|Ai|−1
1· · · X|AN|−1
Nis the term of maximal degree in Q(X)−δ·P(X). But
this contradicts the Nullstellensatz since this polynomial vanishes on QiAi.
If the polynomials are without constant terms, the trivial solution together
with |V| 6= 1 thus imply the existence of a non-trivial solution. It is worth
noting that in the case where Ai=Fqfor all i, the above proof is easily
modified to show a little more, namely that |V| ≡ 0 (mod p) where pis
the characteristic of Fq. This result, which was shown by Warning with an
entirely different method, is known as the Chevalley-Warning theorem [5].
Schanuel showed that a set of congruences fj(a1, . . . , aN)≡0 (mod pνj)
with a prime pand polynomials fj(X1, . . . , XN) over Zwithout constant
terms has a non-trivial solution (a1, . . . , aN)∈ANif Nexceeds the sum
Pjdeg(fj)(pνj−1)(p−1)−1[4]. Here A={x∈Z/pνZ|xp=x}is the set
of so-called Teichm¨uller representatives modulo pν,ν= maxjνj.
The second theorem given here generalizes Schanuel’s result. The last
part of the proof follows, and at one point simplifies, that of Schanuel.
Theorem 2. Consider the polynomials f1(X1, . . . , XN), . . . , fR(X1, . . . , XN)
over Z. Let ν1, . . . , νRbe positive integers, pa prime, and A1, . . . , ANnon-
empty subsets of Zsuch that, for each i, the elements of Aiare pairwise
incongruent modulo p. Assume further
X
i
(|Ai| − 1) >X
j
deg(fj)(pνj−1).
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Then the solution set
V={a∈Y
i
Ai|fj(a)≡0 (mod pνj)for all j}
with the variables restricted to the Aiis not a singleton.
Proof. First assume that all νj= 1. Then Theorem 2 clearly reduces to
Theorem 1 with q=p.
Now let the νjbe arbitrary. We then define, for each i, a polynomial
τi(X)∈Q[X] of degree < p such that τi(a)=(a−ap)/p for every a∈Ai.
Since Aihas at most pelements, such a polynomial can be constructed by
Lagrange interpolation, i.e.
τi(X) = X
a∈Ai

a−ap
p·Y
b∈Ai\{a}
X−b
a−b
.
Recall that a rational number n/m with (n, m) = 1 is called p-adically integral
if p-m, and that such numbers form a subring of Qdenoted Z(p). Since
(a−ap)/p is an integer (by Fermat’s little theorem), and aand bare distinct
modulo p(being distinct elements of Ai), the coefficients of τi(X) are p-
adically integral. Then put σi(X) = Xp+p·τi(X) and note σi(X)∈Z(p)[X],
deg(σi) = p,σi(X)≡Xp(mod p) and σi(a) = afor all a∈Ai.
Next define an operator ∆ : Z(p)[X1, . . . , XN]→Z(p)[X1, . . . , XN] by
letting
(∆f)(X1, . . . , XN)=(f(X1, . . . , XN)p−f(σ1(X1), . . . , σN(XN)))/p.
As in [4], one observes that ∆fhas, in fact, coefficients in Z(p); that deg(∆f)≤
p·deg(f); that ∆c= (cp−c)/p for f=cconstant; that c≡0 (mod pν) if
and only if c, ∆c, . . . , ∆ν−1c≡0 (mod p); and that (∆f)(a) = ∆(f(a)) for
a∈QiAi. For a∈QiAiit is concluded that f(a)≡0 (mod pν) if and only
if (∆if)(a)≡0 (mod p) for all i= 0, . . . , ν −1. Thus one congruence modulo
pνof degree deg(f) can be replaced by νcongruences modulo p, the sum of
whose degrees is at most deg(f)(1 + p+· · · +pν−1) = deg(f)(pν−1)(p−1)−1.
This, together with Theorem 1, finishes the proof.
Theorem 2 is stated only over Z, but it is straightforwardly extended to the
ring of integers in any algebraic number field. The prime pshould then be
replaced by a prime ideal p, and the last assumption by Pi(|Ai| − 1) >
Pjdeg(fj)(qνj−1) where q=N(p) is the norm of p.
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It is a famous result of Olson, answering in part a question of Daven-
port, that a sequence g1, . . . , gNof elements from a finite Abelian p-group
with cyclic factors Z/pνjZhas a non-empty subsequence with sum zero if
its length Nexceeds Pj(pνj−1) [3]. It is remarkable that Olson’s theorem
is equivalent to the special case of Schanuel’s theorem where all fjare of
the form f∗
j(Xp−1
1, . . . , Xp−1
N) with linear f∗
j. Using Theorem 2 with linear fj
instead, one obtains the following extension of Olson’s theorem:
Corollary. Let g1, . . . , gNbe a sequence of elements from a finite Abelian p-
group QjZ/pνjZ. Let A1, . . . , ANbe subsets of Zsuch that each Aicontains
0 and has elements pairwise incongruent modulo p. Assume
X
i
(|Ai| − 1) >X
j
(pνj−1).
Then the equation a1g1+· · ·+aNgN= 0 has a non-trivial solution (a1, . . . , aN)
in QiAi.
References
[1] N. Alon,Combinatorial Nullstellensatz, Combin. Probab. Comput. 8
(1999), 7–29.
[2] C. Chevalley,D´emonstration d’une hypoth`ese de M. Artin, Abh.
Math. Sem. Univ. Hamburg 11 (1936), 73–75.
[3] J. E. Olson,A combinatorial problem on finite abelian groups I, J.
Number Theory 1(1969), 8–10.
[4] S. H. Schanuel,An extension of Chevalley’s theorem to congruences
modulo prime powers, J. Number Theory 6(1974), 284–290.
[5] E. Warning,Bemerkung zur vorstehenden Arbeit von Herrn Chevalley,
Abh. Math. Sem. Univ. Hamburg 11 (1936), 76–83.
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