A Combinatorial Approach to the Groebner Bases for Ideals Generated by Elementary Symmetric Functions

TL;DR

This paper introduces a new combinatorial method to compute Groebner bases for ideals generated by elementary symmetric functions of arbitrary degrees, extending previous specific cases.

Contribution

It develops a generalized approach using symbolic computation to find Groebner bases for ideals generated by elementary symmetric polynomials of any degree.

Findings

Extended Groebner basis computation to arbitrary degrees

Utilized symbolic computation for generalization

Built upon prior specific-degree results

Abstract

Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner basis of the ideal of elementary symmetric polynomials in $n$ variables of degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further generalize their findings by using symbolic computation and experimentation to construct the reduced Groebner basis for the ideal generated by the elementary symmetric polynomials in $n$ variables of arbitrary degrees.