A $p$-adic lower bound for a linear form in logarithms

TL;DR

This paper establishes explicit lower bounds in the p-adic setting for linear forms in p-adic logarithms of rational numbers, advancing tools for Diophantine equations using Padé approximations.

Contribution

It introduces a novel method to derive explicit p-adic lower bounds for linear forms in p-adic logarithms employing Padé approximations of the second kind.

Findings

Derived explicit p-adic lower bounds for linear forms in p-adic logarithms.

Applied Padé approximations to improve bounds in the p-adic context.

Enhanced techniques for solving Diophantine equations using these bounds.

Abstract

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Pad\'e approximations of the second kind.