LCM decomposition of linear differential operators in positive characteristic
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918326129655808 |
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| author | Pagès, Raphaël |
| author_facet | Pagès, Raphaël |
| contents | 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 $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07237 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | LCM decomposition of linear differential operators in positive characteristic Pagès, Raphaël Symbolic Computation Rings and Algebras 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 $L$. |
| title | LCM decomposition of linear differential operators in positive characteristic |
| topic | Symbolic Computation Rings and Algebras |
| url | https://arxiv.org/abs/2602.07237 |