关于具有变量分离性质的p进微分方程
On p-adic differential equations with separation of variables
AI总结：
针对有限域上常微分方程幂级数解计算的p进提升精度需求问题，研究利用微分精度技术得到了变量分离型常微分方程牛顿迭代稳定性的最优界，可应用于多类有限域相关算法并实现显著提速。
摘要：
计算机代数中的若干算法都涉及对给定常微分方程的幂级数解的计算。在有限域上，该问题通常被提升到近似p进环境中以保证适定性。这引发了精度方面的顾虑：我们需要输入具备多高的精度，才能准确计算出输出？在具有变量分离性质的常微分方程的情形下，我们利用最新的微分精度技术，得到了牛顿迭代稳定性的最优界。这些结果可应用于例如有限域上代数数的处理算法、椭圆曲线同源的计算算法，或是有限域中多项式根的确定性求解算法。新得到的界在实际应用中可带来显著的速度提升。
英文摘要：
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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