Desingularization in the $q$-Weyl algebra

TL;DR

This paper develops algorithms for desingularization in the first q-Weyl algebra, enabling the computation of desingularized operators and the q-Weyl closure, with applications to colored Jones polynomials.

Contribution

It introduces an order bound and algorithms for desingularization and q-Weyl closure in the first q-Weyl algebra, advancing computational methods in this area.

Findings

Derived an order bound for desingularized operators.

Presented algorithms for computing desingularized operators and q-Weyl closure.

Applied methods to certify Laurent polynomial sequences in colored Jones polynomials.

Abstract

In this paper, we study the desingularization problem in the first $q$-Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first $q$-Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first $q$-Weyl closure of a given $q$-difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator.