Computing elements of certain form in ideals to prove properties of operators

TL;DR

This paper introduces algorithms for finding specific elements in noncommutative polynomial ideals to prove properties of linear operators, with implementations in Mathematica.

Contribution

It presents novel algorithms for intersecting and computing polynomials in noncommutative ideals, enhancing proof techniques for operator identities.

Findings

Algorithms successfully compute elements in noncommutative ideals.

Implementation in Mathematica package OperatorGB.

Applications demonstrated on operator property proofs.

Abstract

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gr\"obner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the Mathematica package OperatorGB.