Generation of the Symmetric Field by Newton Polynomials in prime Characteristic

Details Generation of the Symmetric Field by Newton Polynomials in prime Characteristic Generation of the Symmetric Field by Newton Polynomials in prime Characteristic

2009-03-18

arxiv.org/abs/0903.3192v2

Mathematics

Commutative Algebra

13 pages

Scientific paper

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.

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