Symmetric polynomials over finite fields

It is shown that two vectors with coordinates in the finite q-element field of characteristic p belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree pk, 2pk, ... , (q - 1)pk, k = 0, 1, 2, ... has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when q = p and the dimension is large compared to p. A relatively small separating set of multisymmetric polynomials over the field of q elements is derived. (c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).