Mathematics > Commutative Algebra
[Submitted on 18 Mar 2009 (v1), last revised 15 Oct 2011 (this version, v2)]
Title:Generation of the Symmetric Field by Newton Polynomials in prime Characteristic
View PDFAbstract: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.
Submission history
From: Maurizio Monge [view email][v1] Wed, 18 Mar 2009 16:13:36 UTC (14 KB)
[v2] Sat, 15 Oct 2011 16:52:48 UTC (14 KB)
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