GENERATION OF THE SYMMETRIC FIELD BY NEWTON POLYNOMIALS IN PRIME CHARACTERISTIC

Let N-m = x(m) + y(m) be the mth Newton polynomial in two variables, for m >= 1. Dvornicich and Zannier proved that in characteristic zero three Newton polynomials N-a, N-b, N-c are always sufficient to generate the symmetric field in x and y, provided that a, b, c are distinct positive integers such that (a, b, c) = 1. In the present paper we prove that in the case of the prime characteristic p the result still holds, if we assume additionally that a, b, c, a - b, a - c, b - c are prime with p. We also provide a counterexample in the case where one of the hypotheses is missing. The result follows from the study of the factorization of a generalized Vandermonde determinant in three variables, which under general hypotheses factors as the product of a trivial Vandermonde factor and an irreducible factor. On the other side, the counterexample is connected to certain cases where Schur polynomials factor as a product of linear factors.