ArticlePDF Available

# Chevalley’s theorem with restricted variables

Authors:

## 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.
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 AiK.
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 AiKif
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
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
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)(q1).
1
Then the solution set
V={aY
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)q1) satisﬁes
P(x) = (1 for x=a,
0 for xQiAi\{a}.
Further, Q(X) = QiQbAi\{ai}(Xib) satisﬁes
Q(x) = (δfor x=a,
0 for xQiAi\{a},
with some (non-zero) δFq. 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)(pνj1)(p1)1[4]. Here A={xZ/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)(pνj1).
2
Then the solution set
V={aY
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)=(aap)/p for every aAi.
Since Aihas at most pelements, such a polynomial can be constructed by
Lagrange interpolation, i.e.
τi(X) = X
aAi
aap
p·Y
bAi\{a}
Xb
ab
.
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
(aap)/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 aAi.
Next deﬁne an operator ∆ : Z(p)[X1, . . . , XN]Z(p)[X1, . . . , XN] by
letting
(∆f)(X1, . . . , XN)=(f(X1, . . . , XN)pf(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= (cpc)/p for f=cconstant; that c0 (mod pν) if
and only if c, c, . . . , ν1c0 (mod p); and that (∆f)(a) = ∆(f(a)) for
aQiAi. For aQiAiit 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+· · · +pν1) = deg(f)(pν1)(p1)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)(qνj1) where q=N(p) is the norm of p.
3
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(pνj1) [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(Xp1
1, . . . , Xp1
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
(pνj1).
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,emonstration d’une hypothese 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.
4
... Katz generalized Chevalley's Theorem to a wider class of finite commutative rings [31,23,20]. U. Schauz and D. Brink considered solutions lying in rectangular subsets of F N [32,7]. In some special cases the degree can be replaced by the p-weight degree [8]. ...
... Section 13 contains some Chevalley-Warning type results for functions with restricted domain. In the special case that the domain is restricted to a subgroup of F N we obtain a p-weight degree version of the Schauz-Brink Theorem [32], [7,Theorem 1], and we also obtain that the number of solutions in the subgroup is divisible by the field characteristic (Theorem 13.2). ...
... We start from a generalization of Chevalley's Theorem given in [32]. In [7], this result was stated in the following form: ...
Article
Full-text available
We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.
... Katz generalized Chevalley's Theorem to a wider class of finite commutative rings [Sch74,Kat09]. D. Brink considered solutions lying in rectangular subsets of F N ( [Bri11], with a p-weight degree version given in [CGM19]). Brink's result was used in [KS18,Aic19] to solve equations over finite nilpotent rings, groups, and generalizations of these structures. ...
... This allows us to generalize Asgarli's proof of Warning's Second Theorem [Asg18] to derive its p-weight degree improvement [MM95, Theorem 2] (Theorem 14.1). A similar improvement of Brink's Theorem [Bri11, Theorem 1] can be obtained in the special case that the domain is restricted to a subgroup of F N (Section 13) and we also obtain that the number of solutions in the subgroup is divisible by the field characteristic (Theorem 13.2). Warning's First Theorem can be strenghtened if we know that the functions are not surjective: such "restricted range" versions are given in Theorem 13.5 and Corollary 13.6. ...
... We start from a variant of Chevalley's Theorem that was formulated and proved in [Bri11] in the following form: ...
Preprint
Full-text available
We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.
... A similar argument gives the following result of Schauz [Sc08] and Brink [Br11]. Theorem 1.3. ...
... In fact [CFS17, Thm. 1.6] is a further ring theoretic generalization motivated by work of Brink[Br11], but in the present paper we will only consider polynomials over a field. ...
Preprint
Full-text available
We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde subsets of $\F_q$ defined by G\'acs-Weiner and Sziklai-Tak\'ats. We then define an invariant $\uomega(X)$ of a nonempty subset of $\F_q^n$. Our second result involves $X$-restricted variables when the degrees of the polynomials are small compared to $\uomega(X)$. We end by exploring various classes of subsets for which $\uomega(X)$ can be bounded from below.
... In the case K = Q we are studying systems of congruence modulo (varying) powers of a (fixed) prime p. Second, we study solutions in which each variable is independently restricted to a finite subset of Z K satisfying the condition that no two distinct elements are congruent modulo p. Theorem 1.2 extends work of Schanuel [27], Baker-Schmidt [9], Schauz [28], Wilson [32] and Brink [8]. These works are largely motivated by applications to combinatorics. ...
... , Brink[8]) Let G ∼ = r i=1 Z/ p v i be a p-group of exponent p v . Let n ∈ Z + ,and let A = (A 1 , . . . ...
Article
Full-text available
We present a generalization of Warning’s second theorem to polynomial systems over a finite local principal ring with restricted input and relaxed output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
... , X n ) = 0, 1 ≤ i ≤ k, has a solution in F n q , then it has another solution. Later on, Brink [3] and Schauz [13] used Alon's ideas to prove a restricted variable version of this theorem. ...
... We write |C| for the cardinality of a finite set C. The following result is a restricted variable version of the Chevalley-Warning theorem (for a proof, see Brink [3] or Schauz [13]). ...
Article
Full-text available
We show that a weaker version of the well-known theorem of Morlaye and Joly on diagonal equations is a simple consequence of a restricted variable version of the Chevalley-Warning theorem. Moreover, we extend the result of Morlaye and Joly to the case of an equation of the form b_1D_{m_1}(X_1,a_1)+...+b_nD_{m_n}(X_n,a_n)=c, where D_{m_1}(X_1,a_1),...,D_{m_n}(X_n,a_n) are Dickson polynomials.
... Theorem 2.18 (U. Schauz [23], D. Brink [6]). Let P 1 (t 1 , . . . ...
Preprint
Full-text available
We generalize the notion of Erd\H{o}s-Ginzburg-Ziv constants -- along the same lines we generalized in earlier work the notion of Davenport constants -- to a `higher degree" and obtain various lower and upper bounds. These bounds are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. We also consider to what extent a theorem due independently to W.D. Gao and the first author that relates these two parameters extends to this higher degree setting. Two simple examples that capture the essence of these higher degree Erd\H{o}s-Ginzburg-Ziv constants are the following. 1) Let $\nu_p(m)$ denote the $p-$adic valuation of the integer $m$. Suppose we have integers $t | {m \choose 2}$ and $n=t+2^{\nu_2(m)}$, then every sequence $S$ over ${\mathbb Z}_2$ of length $|S| \geq n$ contains a subsequence $S'$ of length $t$ for which $\sum_{a_{i_1},\ldots, a_{i_m} \in S'} a_{i_1}\cdots a_{i_m} \equiv 0 \pmod{2}$, and this is sharp. 2) Suppose $k=3^{\alpha}$ for some integer $\alpha \geq 2$. Then every sequence $S$ over ${\mathbb Z}_3$ of length $|S| \geq k+6$ contains a subsequence $S'$ of length $k$ for which $\sum_{a_i, a_j, a_h \in S'} a_ia_ja_k \equiv 0 \pmod{3}$, and this is sharp. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions (symmetric polynomials) have to vanish on the elements of a subsequence of prescribed length. The Erd\H{o}s-Ginzburg-Ziv Theorem is just the case where a sequence of length $2n-1$ over ${\mathbb Z}_n$ contains a subsequence $S'=(a_1, \ldots, a_n)$ of length $n$ that vanishes when substituted in the linear symmetric polynomial $a_1+\cdots+a_n. ... We first need the following variation of [Bri11, Theorem 1] and [KS18, Theorem 3.2], which is proved using several arguments from the proof of [Alo99, Theorem 3.1] and from [Bri11]. ... Preprint Recently, M. Kompatscher proved that for each finite supernilpotent algebra$\mathbf{A}$in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let$\mu$be the maximal arity of the fundamental operations of$\mathbf{A}$, and let $d := |A|^{\log_2 (\mu) + \log_2 (|A|) + 1}.$ Applying a method that G. K\'{a}rolyi and C. Szab\'{o} had used to solve equations over finite nilpotent rings, we show that for$\mathbf{A}$, there is$c \in \mathbb{N}$such that a solution of every system of$s$equations in$n$variables can be found by testing at most$c n^{sd}$(instead of all$|A|^n\$ possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.
... Our proof depends on the following result of Brink [2], which can be viewed as a generalization of Chevalley's well-known theorem [4] and its somewhat forgotten extension by Schanuel [14]. ...
Preprint
Full-text available
We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics, which may be of independent interest.
... a) It follows from (2) that #V X = 1. This is the Restricted Variable Chevalley Theorem of Schauz [Sc08a] and Brink [Br11]. b) Take X i = F q for all i, and suppose (1) holds, i.e., r j=1 deg(P j ) < n. ...
Article
Inspired by recent work of I. Baoulina, we give a simultaneous generalization of the theorems of Chevalley-Warning and Morlay
Article
Recently Schauz and Brink independently extended Chevalley's theorem to polynomials with restricted variables. In this note we give an improvement to Schauz-Brink's theorem via the ground field method. The improvement is significant in the cases where the degree of the polynomial is large compared to the weight of the degree of the polynomial.
Article
This paper continues the discussion of the number s(G) defined, for a finite Abelian group G, as the least number s such that an arbitrary sequence of length s of group elements has a subsequence whose product is 1.
Article
Olson determined, for each finite abelian p-group G, the maximal length of a sequence of elements of G such that no subsequence has zero sum, thus settling (at least for these groups) a problem raised by Davenport in connection with factorization in number fields. This problem is equivalent to one on simultaneous linear congruences to which one seeks solutions with the variables restricted to the values 0 and 1. In the present note, the analogous problem for forms of arbitrary degree is settled, again with best possible results. The main tool is an extension of Chevalley's theorem on finite fields to congruences modulo prime powers. This in turn is deduced from Chevalley's theorem by a simple device which circumvents the use of Witt vectors.
Article
If G is a finite Abelian group, for what number s is it true that an arbitrary sequence of length s of group elements has a subsequence whose product is 1? This question is answered for p-groups.
Dé d'une hypoth ese de M
• C Chevalley
C. Chevalley: Dé d'une hypoth ese de M. Artin, Abh. Math. Sem. Univ. Hamburg 11 (1936), 73–75.
• N Alon
N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7-29.