Addition of sets via symmetric polynomials — A polynomial method

Let A 1 , … , A h be finite non-empty subsets of a field K and let s k ( x 1 , … , 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 × ⋯ × A h by the polynomial s k , with and without the restriction that the elements of the h -tuples are pairwise distincts.