A Factorization Algorithm for G-Algebras and Applications

TL;DR

This paper introduces a new algorithm for factorizing elements in G-algebras, leveraging their finite factorization domain property, and extends this to a factorized Gr"obner basis algorithm for solving systems of equations with polynomial coefficients.

Contribution

The paper presents the first algorithm for all factorizations in G-algebras and adapts the Gr"obner basis method for these algebras, enabling advanced analysis of functional equations.

Findings

Algorithm successfully finds all factorizations in G-algebras.

Extension of Gr"obner basis method to factorized form for G-algebras.

Applicable to solving systems of linear partial functional equations.

Abstract

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is any $G$-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gr\"obner basis algorithm for $G$-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from $\mathcal{G}$. Additionally, it is possible to include inequality constraints for ideals in the input.