Solving $p$-adic polynomial equations using Jarratt's Method

TL;DR

This paper introduces a simplified iterative method for solving polynomial equations over the $p$-adic numbers, demonstrating higher convergence order and relaxed initial conditions compared to previous methods.

Contribution

It presents a new $p$-adic iterative root-finding method with improved convergence and less restrictive initial conditions than prior approaches.

Findings

Higher order of convergence than previous $p$-adic methods

Allows starting with multiple roots of congruences

Demonstrates effectiveness through implementation

Abstract

We implement an iterative numerical method to solve polynomial equations $f(x)=0$ in the $p$-adic numbers, where $f(x) \in\mathbb{Z}_p[x]$. This method is a simplified $p$-adic analogue of Jarratt's method for finding roots of functions over the real numbers. We establish that our method has a higher order of convergence than J.F.T. Rabago's $p$-adic version of Olver's method from 2016. Moreover, we weaken the initial conditions in Rabago's method, which allows us to start the iteration with a multiple root of the congruence $f(x) \equiv 0 \bmod{p}$.