Linear extension of the Erdos-Heilbronn conjecture
Abstract
The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements a1,...,an of F, the set a1x1+...+anxn: x1,....,xn are distinct elements of A has cardinality at least minp(F)-delta, n(|A|-n)+1, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and a1+a2=0. In this paper we prove this conjecture of Sun when p(F)≥ n(3n-5)/2. We also obtain a sharp lower bound for the cardinality of the restricted sumset x1+...+xn: x1∈ A1,...,xn∈ An, and P(x1,...,xn)=0, where A1,...,An are finite subsets of a field F and P(x1,...,xn) is a general polynomial over F.