On value sets of polynomials over a field

• Authors: Zhi-Wei Sun
• Venue: Finite Fields and Their Applications
• Year: 2008

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 letf(x"1,...,x"n)=a"1x"1^k+...+a"nx"n^k+g(x"1,...,x"n)@?F[x"1,...,x"n] with k@?{1,2,3,...}, a"1,...,a"n@?F@?{0} and degg=min{p(F),@?i=1n@?|A"i|-1k@?+1}. When k=n and |A"i|=i for i=1,...,n, we also have|{f(x"1,...,x"n):x"1@?A"1,...,x"n@?A"n,and x"ix"j if ij}|=min{p(F),@?i=1n@?|A"i|-ik@?+1}; consequently, if n=k then for any finite subset A of F we have|{f(x"1,...,x"n):x"1,...,x"n@?A, and x"ix"j if ij}|=min{p(F),|A|-n+1}. In the case nk, we propose a further conjecture which extends the Erdos-Heilbronn conjecture in a new direction.