Regular sequences of power sums and complete symmetric polynomials

TL;DR

This paper investigates regular sequences of symmetric polynomials, especially power sums, in polynomial rings with three and four variables, proposing conjectures and providing partial proofs for their regularity.

Contribution

It establishes that pairs of power sums always form regular sequences and proposes a conjecture for triples in four variables, supporting it with partial evidence and extending known results.

Findings

Any two power sums in three or more variables form a regular sequence.

A conjecture is proposed for when three power sums form a regular sequence in four variables.

Sequences of consecutive power sums with certain parameters form regular sequences in n variables.

Abstract

In this article, we carry out the investigation for regular sequences of symmetric polynomials in the polynomial ring in three and four variable. Any two power sum element in $\mathbb{C}[x_1,x_2,...,x_n]$ for $n \geq 3$ always form a regular sequence and we state the conjecture when $p_a,p_b,p_c$ for given positive integers $a<b<c$ forms a regular sequence in $\mathbb{C}[x_1,x_2,x_3,x_4]$. We also provide evidence for this conjecture by proving it in special instances. We also prove that any sequence of power sums of the form $p_{a}, p_{a+1},..., p_{a+ m-1},p_b$ with $m <n-1$ forms a regular sequence in $\mathbb{C}[x_1,x_2,...,x_n]$. We also provide partial evidence in support of conjecture's given by Conca, Krattenthaler and Watanabe on regular sequences of symmetric polynomials.