# Computations with p-adic numbers

## TL;DR

This paper discusses algorithms for p-adic numbers, compares different implementations, and introduces a framework for analyzing precision issues in p-adic computations.

## Contribution

It presents various implementations of p-adic numbers and a general framework for studying and managing precision errors in p-adic algorithms.

## Key findings

- Different implementations of p-adic numbers are compared.
- A framework for analyzing precision issues is introduced.
- Applications demonstrate the framework's effectiveness.

## Abstract

This document contains the notes of a lecture I gave at the "Journ\'ees Nationales du Calcul Formel" (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of p-adic numbers and compare them and second, we introduce a general framework for studying precision issues and apply it in several concrete situations.