On value sets of polynomials over a field

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