On p-adic differential equations with separation of variables

Pierre Lairez (1) , Tristan Vaccon (2)

1 Institut für Mathematik [Berlin]
2 Rikkyo University [Tokyo]

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 equation with separation of variables, we apply the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. It applies, 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.

Mots clés: Ordinary differential equation; p-adic numbers; Newton iteration; numerical stability; differential precision

Domaines: Calcul formel [cs.SC]

Pierre Lairez, Tristan Vaccon. On p-adic differential equations with separation of variables. 2016. ⟨hal-01265226v1⟩