Olver's Method for Approximating Roots of p-Adic Polynomials Equations

TL;DR

This paper adapts Olver's root-finding method to p-adic polynomial equations, providing a new approach for approximating roots within the p-adic integer framework.

Contribution

It introduces an analogue of Olver's method specifically designed for solving polynomial equations over p-adic integers, extending classical techniques to p-adic analysis.

Findings

Developed a p-adic Olver's method for root approximation

Proved convergence properties of the method in $ ext{Z}_p$

Demonstrated effectiveness through theoretical analysis

Abstract

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The solution is approached through an analogue of Olver's method for finding roots of polynomial equations $P(x)=0$ in $\mathbb{Z}_p$.