Prime ideals and regular sequences of symmetric polynomials

TL;DR

This paper characterizes subsets of power sum symmetric polynomials that generate prime ideals or form regular sequences in a polynomial ring, using algebraic and number-theoretic techniques.

Contribution

It provides a comprehensive description of when sets of power sum symmetric polynomials generate prime ideals or are regular sequences, introducing new families of prime ideals and regular sequences.

Findings

Identifies conditions for prime ideal generation by power sum polynomials.

Constructs large families of prime ideals using Serre's criterion and roots of unity.

Deduces new results on regular sequences of symmetric polynomials.

Abstract

Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with $a \in A$ generate a prime ideal in S or the set of polynomials $p_a$ with $a \in A$ is a regular sequence in S. We produce a large families of prime ideals by exploiting Serre's criterion for normality [4, Theorem 18.15] with the help of arithmetic considerations, vanishing sums of roots of unity [9]. We also deduce several other results concerning regular sequences of symmetric polynomials.