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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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