Chevalley’s theorem with restricted variables

Abstract

First, a generalization of Chevalley’s classical theorem from 1936 on polynomial equations f(x 1,...,x N ) = 0 over a finite field K is given, where the variables x i are restricted to arbitrary subsets A i ⊆ K. The proof uses Alon’s Nullstellensatz. Next, a theorem on integer polynomial congruences f(x 1,...,x N ) ≡ 0 (mod p v ) 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.