A new extension of the Erdos-Heilbronn conjecture

TL;DR

This paper extends the Erdos-Heilbronn conjecture by establishing a new lower bound on the size of a polynomial value set over finite subsets of a field, considering constraints on the variables.

Contribution

It introduces a novel lower bound for polynomial value sets that generalizes the Erdos-Heilbronn conjecture to multivariate polynomials with specific degree conditions.

Findings

Provides a new lower bound for polynomial value set cardinality

Extends the Erdos-Heilbronn conjecture to multivariate polynomials

Incorporates variable restrictions into the bound

Abstract

Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {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}. The result extends the Erdos-Heilbronn conjecture in a new way.