Addition of sets via symmetric polynomials — A polynomial method

Let A1,…,AhA1,…,Ah be finite non-empty subsets of a field KK and let sk(x1,…,xh)sk(x1,…,xh) be the elementary symmetric polynomial of degree kk in hh indeterminates. Here we present some estimates for the cardinality of the sets of the images of all hh-tuples of A1×⋯×AhA1×⋯×Ah by the polynomial sksk, with and without the restriction that the elements of the hh-tuples are pairwise distincts.