LCM decomposition of linear differential operators in positive characteristic

TL;DR

This paper introduces a polynomial-time algorithm for computing the least common left multiple (LCLM) decomposition of linear differential operators over fields of positive characteristic, leveraging Frobenius normal form and module isomorphisms.

Contribution

The paper presents the first efficient algorithm for LCLM decomposition of differential operators in positive characteristic, combining Frobenius form analysis and module isomorphisms.

Findings

Algorithm runs in polynomial time in order, degree, and characteristic p.

Shape of factorization derived from Frobenius normal form of p-curvature.

Constructs LCLM decomposition via module isomorphism techniques.

Abstract

We present an algorithm to compute $\mathrm{LCLM}$-decompositions for linear differentials operators with coefficients in the rational function field of characteristic $p$, $\mathbb{F}_{p^n}(t)$. We show that for an operator $L$ of order $r$ with coefficients of degree $d$, it finishes in polynomial time in $r$, $d$ and $p$. This algorithm proceeds in three steps. We begin by showing that the ''shape'' of the factorisation of $L$ can be easily obtained from the Frobenius normal form of its $p$-curvature, which can be efficiently computed an algorithm from Bostan, Caruso and Schost. Using results from the thesis of the author, we are then able to construct an operator $L^*$ in the same equivalence class as $L$ for which an $\mathrm{LCLM}$-decomposition is known. Finally, by computing an isomorphism between the quotient modules $\mathbb{F}_q(t)\langle\partial\rangle/\mathbb{F}_q(t)\langle\partial\rangle L^*$ and $\mathbb{F}_q(t)\langle\partial\rangle/\mathbb{F}_q(t)\langle\partial\rangle L$, we find a corresponding $\mathrm{LCLM}$-decomposition of $L$.