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.
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References
N. Alon: Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7–29.
C. Chevalley: Démonstration d’une hypothèse de M. Artin, Abh. Math. Sem. Univ. Hamburg 11 (1936), 73–75.
J. E. Olson: A combinatorial problem on finite abelian groups I, J. Number Theory 1 (1969), 8–10.
S. H. Schanuel: An extension of Chevalley’s theorem to congruences modulo prime powers, J. Number Theory 6 (1974), 284–290.
E. Warning: Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Univ. Hamburg 11 (1936), 76–83.