# On p-adic differential equations with separation of variables

Abstract : Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. The results apply, for example, to algorithms for manipulating algebraic numbers over finite fields, for computing isogenies between elliptic curves or for deterministically finding roots of polynomials in finite fields. The new bounds lead to significant speedups in practice.
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Type de document :
Communication dans un congrès
ISSAC 2016, Jul 2016, Waterloo, Ontario, Canada. Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation, 〈http://issac-symposium.org/2016/〉. 〈10.1145/2930889.2930912〉
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https://hal.archives-ouvertes.fr/hal-01265226
Contributeur : Pierre Lairez <>
Soumis le : mardi 10 mai 2016 - 09:25:06
Dernière modification le : samedi 16 décembre 2017 - 07:18:04
Document(s) archivé(s) le : mardi 15 novembre 2016 - 19:37:01

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Pierre Lairez, Tristan Vaccon. On p-adic differential equations with separation of variables. ISSAC 2016, Jul 2016, Waterloo, Ontario, Canada. Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation, 〈http://issac-symposium.org/2016/〉. 〈10.1145/2930889.2930912〉. 〈hal-01265226v2〉

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