On p-Adic Differential Equations with Separation of Variables

Pierre Lairez, Tristan Vaccon
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引用次数: 13

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.
关于分离变量的p进微分方程
计算机代数中的几种算法涉及给定常微分方程的幂级数解的计算。在有限域上,这个问题通常在近似的$p$进设置中被提升为适定的。这引起了对精度的关注:输入需要多少精度才能准确地计算输出?对于具有分离变量的常微分方程,我们利用最新的微分精度技术得到牛顿迭代稳定性的最优界。例如,这些结果适用于在有限域上操作代数数的算法,用于计算椭圆曲线之间的同质性,或用于确定地在有限域中查找多项式的根。新的边界在实践中导致了显著的加速。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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