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 $$f(x_1,...,x_n)=a_1{x}_1^k+...+a_n{x}_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n]$$ with k in {1,2,3,...}, a_1,...,a_n in F\{0} and deg(g)<k. We show that $$|f(x_1,...,x_n):x_{1}in{A}_1,...,x_{n}in{A}_n|\ge minp(F),\sum _{i=1}^n[(|A_i|-1)/k]+1.$$ When $k\ge n$ and $|A_i|\ge i$ for $i=1,...,n$, we also have $$|f(x_1,...,x_n):x_{1}in{A}_1,...,x_{n}in{A}_n,and{x}_{i}not=x_{j}ifinot=j|\ge minp(F),\sum _{i=1}^n[(|A_i|-i)/k]+1;$$ consequently, if $n\ge k$ then for any finite subset A of F we have $$|f(x_1,...,x_n):x_1,...,x_{n}inA,and{x}_{i}not=x_{j}ifinot=j|\ge minp(F),|A|-n+1.$$ In the case $n>k$ we propose a further conjecture which extends the Erdos-Heilbronn conjecture in a new direction.