On p-adic differential equations with separation of variables

TL;DR

This paper develops optimal bounds on the precision needed for p-adic solutions of differential equations with separation of variables, improving computational stability and efficiency in algebraic algorithms over finite fields.

Contribution

It introduces the use of differential precision to analyze stability and precision requirements for p-adic differential equations with separation of variables, leading to practical speedups.

Findings

Optimal bounds on input precision for p-adic differential equations

Enhanced stability of Newton iteration in this context

Significant practical speedups in algebraic computations

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.