An Algorithm for Finding Symmetric Gr\"obner Bases in Infinite Dimensional Rings

TL;DR

This paper introduces an explicit algorithm for computing Gr"obner bases of symmetric ideals in infinite-dimensional polynomial rings, enabling symbolic computation and solving the ideal membership problem in this context.

Contribution

The paper presents the first explicit algorithm for finding Gr"obner bases of symmetric ideals in infinite-dimensional rings, advancing symbolic computation in these rings.

Findings

Algorithm successfully computes Gr"obner bases for symmetric ideals.

Solves the ideal membership problem for symmetric ideals in infinite rings.

Enables new symbolic computation capabilities in infinite-dimensional polynomial rings.

Abstract

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.