Addition of sets via symmetric polynomials — A polynomial method

Abstract

Let $A_1,\dots ,A_h$ be finite non-empty subsets of a field $K$ and let $s_k(x_1,\dots ,x_h)$ be the elementary symmetric polynomial of degree $k$ in $h$ indeterminates. Here we present some estimates for the cardinality of the sets of the images of all $h$-tuples of $A_1\times \cdots \times A_h$ by the polynomial $s_k$, with and without the restriction that the elements of the $h$-tuples are pairwise distincts.