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 $L$.

翻译：我们提出一种算法，用于计算系数在正特征$p$的有理函数域$\mathbb{F}_{p^n}(t)$中的线性微分算子的$\mathrm{LCLM}$分解。我们证明，对于阶数为$r$、系数次数为$d$的算子$L$，该算法可在$r$、$d$和$p$的多项式时间内完成。该算法包含三个步骤。首先，我们说明$L$的分解"形式"可以其$p$-曲率的Frobenius标准型直接获得，而该标准型可通过Bostan、Caruso和Schost的算法高效计算。利用作者学位论文中的结果，我们随后能够构造一个与$L$处于同一等价类且已知其$\mathrm{LCLM}$分解的算子$L^*$。最后，通过计算商模$\mathbb{F}_q(t)\langle\partial\rangle/\mathbb{F}_q(t)\langle\partial\rangle L^*$与$\mathbb{F}_q(t)\langle\partial\rangle/\mathbb{F}_q(t)\langle\partial\rangle L$之间的同构，我们得到$L$对应的$\mathrm{LCLM}$分解。