Generation of the Symmetric Field by Newton Polynomials in prime Characteristic

TL;DR

This paper extends a known result about generating symmetric fields with Newton polynomials from characteristic zero to prime characteristic, under specific coprimality conditions, and explores related polynomial factorizations.

Contribution

It proves that in prime characteristic, three Newton polynomials generate the symmetric field if certain coprimality conditions are met, extending previous characteristic zero results.

Findings

The result holds in prime characteristic under coprimality conditions.

Counterexamples are provided when conditions are not met.

Factorization of Vandermonde determinants is key to the proof.

Abstract

Let $N_m = x^m + y^m$ be the $m$-th Newton polynomial in two variables, for $m \geq 1$. 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 case of 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, that 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 the Schur polynomials factor as a product of linear factors.