Effective Approximation and Diophantine Applications

TL;DR

This paper uses the Thue-Siegel method to improve bounds on the irrationality measure of certain algebraic numbers and applies these results to bound solutions of related Diophantine equations.

Contribution

It provides effective improvements on Liouville's irrationality measure for specific algebraic numbers and derives bounds on solutions to related Diophantine equations.

Findings

Improved bounds on irrationality measures for certain algebraic numbers.

Polynomial bounds on the size of solutions depending on parameter a.

Uniform bounds on the number of solutions, independent of parameter a.

Abstract

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some corresponding Diophantine equations. We obtain bounds for the size of solutions, which depend polynomially on $a$, and bounds for the number of these solutions, which are independent of $a$ and in some cases even independent of the degree of the equation.