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.

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References

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Correspondence to David Brink.

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Brink, D. Chevalley’s theorem with restricted variables. Combinatorica 31, 127 (2011). https://doi.org/10.1007/s00493-011-2504-z

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Mathematics Subject Classification (2000)

  • 11D79
  • 20K01