LCM decomposition of linear differential operators in positive characteristic

2026/02/06 by Raphaël Pagès · 1 voice
Computer Science · Mathematics · #cs.SC #math.RA

paper · pdf · doi:10.48550/arxiv.2602.07237

Abstract

We present an algorithm to compute LCLM-decompositions for linear differentials operators with coefficients in the rational function field of characteristic p, \mathbbF_pⁿ(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 LCLM-decomposition is known. Finally, by computing an isomorphism between the quotient modules \mathbbF_q(t)⟨∂⟩/\mathbbF_q(t)⟨∂⟩ L^* and \mathbbF_q(t)⟨∂⟩/\mathbbF_q(t)⟨∂⟩ L, we find a corresponding LCLM-decomposition of L.

Citations

• A unified approach for degree bound estimates of linear differential operators
• Symbolic-Numeric Factorization of Differential Operators
• Fast computation of hyperelliptic curve isogenies in odd characteristic
• Computation of the Similarity Class of the p-Curvature
• Computing the bound of an Ore polynomial. Applications to factorization
• Power Series Solutions of Singular (q)-Differential Equations
• Rings and Categories of Modules

Discussions

• Rapha\"el Pag\`es: LCM decomposition of linear differential operators in positive characteristic https://arxiv.org/abs/2602.07237 https://arxiv.org/pdf/2602.07237 https://arxiv.org/html/2602.07237 [bsky, 0 points, 0 comments]

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