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Elsevier, Finite Fields and Their Applications, 2(14), p. 470-481, 2008

DOI: 10.1016/j.ffa.2007.05.002

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On value sets of polynomials over a field

Journal article published in 2007 by Zhi-Wei Sun
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A_1,...,A_n be finite nonempty subsets of F, and let $f(x_1,...,x_n)=a_1x_1^k+...+a_nx_n^k+g(x_1,...,x_n)𝟄 F[x_1,...,x_n]$ with k in {1,2,3,...}, a_1,...,a_n in F\{0} and deg(g)<k. We show that $|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n}| ≥ min{p(F),∑_{i=1}^n[(|A_i|-1)/k]+1}.$ When $k≥ n$ and $|A_i|≥ i$ for $i=1,...,n$, we also have $|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n, and x_i not=x_j if i not=j}| ≥ min{p(F),∑_{i=1}^n[(|A_i|-i)/k]+1};$ consequently, if $n≥ k$ then for any finite subset A of F we have $|{f(x_1,...,x_n): x_1,...,x_n in A, and x_i not=x_j if i not=j}| ≥ min{p(F),|A|-n+1}.$ In the case $n>k$ we propose a further conjecture which extends the Erdos-Heilbronn conjecture in a new direction.