🤖 AI Summary
This work addresses the efficient computation of least common left multiple (LCLM) factorizations for linear differential operators over rational function fields in positive characteristic. The authors propose a three-step algorithm: first, the structure of factors is determined via the Frobenius normal form of the $p$-curvature; second, equivalent operators are constructed; and third, the LCLM decomposition is obtained by computing module isomorphisms of quotient modules. By integrating $p$-curvature analysis, the Bostan–Caruso–Schost algorithm, operator equivalence theory, and module isomorphism techniques, this approach achieves polynomial time complexity for the first time in terms of the operator order $r$, the degree $d$ of the coefficients, and the characteristic $p$, thereby overcoming a longstanding computational bottleneck in LCLM factorization in positive characteristic settings.
📝 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 $L$.