Desingularization and p-Curvature of Recurrence Operators

TL;DR

This paper explores the relationship between the $p$-curvature of linear recurrence operators and their singularities, introducing faster desingularization methods to improve factorization and computation of $p$-curvature in characteristic $p$.

Contribution

It establishes a novel relation between the characteristic polynomial of $p$-curvature and true singularities, and develops faster algorithms for desingularization and computing $p$-curvature.

Findings

Established a link between $ ext{chi}(L)$ and singularities.

Developed a faster desingularization algorithm.

Improved efficiency of computing $p$-curvature.

Abstract

Linear recurrence operators in characteristic $p$ are classified by their $p$-curvature. For a recurrence operator $L$, denote by $\chi(L)$ the characteristic polynomial of its $p$-curvature. We can obtain information about the factorization of $L$ by factoring $\chi(L)$. The main theorem of this paper gives an unexpected relation between $\chi(L)$ and the true singularities of $L$. An application is to speed up a fast algorithm for computing $\chi(L)$ by desingularizing $L$ first. Another contribution of this paper is faster desingularization.