ZpL: a p-adic precision package

TL;DR

The paper introduces ZpL, a new p-adic precision package for mathematical software, utilizing automatic differentiation for sharp precision tracking, with demonstrated advantages in linear algebra, algebra, and differential equations.

Contribution

It presents a novel p-adic precision package based on ultrametric precision theory and automatic differentiation, improving accuracy and efficiency over previous implementations.

Findings

Enhanced precision tracking in p-adic computations

Complexity analysis of the algorithms

Empirical improvements demonstrated in various mathematical applications

Abstract

We present a new package ZpL for the mathematical software system SM. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in [4]. The underlying algorithms are mostly based on automatic dierentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the bene-ts of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, com-mutative algebra and dierential equations.