Contraction of Ore Ideals with Applications

TL;DR

This paper introduces an algorithm to compute contraction ideals of Ore operators, enabling the construction of desingularized operators with minimal degree and content, useful for applications like sequence certification and conjecture verification.

Contribution

It presents a novel algorithm for contraction ideal computation in Ore algebras, extending desingularization techniques to polynomial coefficient operators.

Findings

Algorithm successfully computes contraction ideals.

Enables construction of minimal-degree, minimal-content desingularized operators.

Applications include certifying integer sequences and verifying mathematical conjectures.

Abstract

Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator $L$ with polynomial coefficients in $x$, it generates a left ideal $I$ in the Ore algebra over the field $\mathbf{k}(x)$ of rational functions. We present an algorithm for computing a basis of the contraction ideal of $I$ in the Ore algebra over the ring $R[x]$ of polynomials, where $R$ may be either $\mathbf{k}$ or a domain with $\mathbf{k}$ as its fraction field. This algorithm is based on recent work on desingularization for Ore operators by Chen, Jaroschek, Kauers and Singer. Using a basis of the contraction ideal, we compute a completely desingularized operator for $L$ whose leading coefficient not only has minimal degree in $x$ but also has minimal content. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler.