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Raphaël Pagès

Publications and source records attributed to Raphaël Pagès.

4 recordsLinked to original sources
arXiv2026

Factoring Linear Differential Operators in Positive Characteristic by means of Solving a Norm Equation

The solutions of the equation $f^{(p-1)} + f^p = h^p$ in the unknown function $f $over an algebraic function field of characteristic $p$ are very closely linked to the structure and factorisations of linear differential operators with coefficients in function fields of characteristic $p$. However, while being able to solve this equation over general algebraic function fields is necessary even for operators with rational coefficients, no general resolution method has been developed. We present an algorithm for testing the existence of solutions in polynomial time in the ``size'' of h and an algorithm based on the computation of Riemann-Roch spaces and the selection of elements in the divisor class group, for computing solutions of size polynomial in the ``size'' of h in polynomial time in the size of h and linear in the characteristic $p$, and discuss its applications to the factorisation of linear differential operators in positive characteristic $p$.

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arXiv2026

Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

We design a fast algorithm that computes, for a given linear differential operator with coefficients in $Z[x ]$, all the characteristic polynomials of its p-curvatures, for all primes $p < N$ , in asymptotically quasi-linear bit complexity in N. We discuss implementations and applications of our algorithm. We shall see in particular that the good performances of our algorithm are quickly visible.

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arXiv2026

LCM decomposition of linear differential operators in positive characteristic

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 $L$.

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