Fast evaluation of some p-adic transcendental functions

TL;DR

This paper introduces efficient algorithms for computing various p-adic elementary and special functions with quasi-linear complexity, adapting binary splitting and bit-burst strategies to the p-adic setting.

Contribution

It presents novel algorithms for p-adic functions that achieve quasi-linear complexity, extending classical methods to the p-adic context.

Findings

Algorithms for p-adic functions with quasi-linear complexity

Adaptation of binary splitting and bit-burst strategies to p-adic setting

Efficient computation of p-adic logarithms, exponentials, polylogarithms, and hypergeometric functions

Abstract

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.