Hermite-Thue equation: Pad\'e approximations and Siegel's lemma

TL;DR

This paper explores the use of Padé approximations and Siegel's lemma to improve Diophantine approximation measures, focusing on the exponential function and the structure of Hermite-Padé approximations.

Contribution

It demonstrates the existence of a large common factor among minors of the coefficient matrix, enabling improved bounds via Siegel's lemma for exponential function approximations.

Findings

Existence of a large common factor among minors of the matrix.

Application of Bombieri-Vaaler Siegel's lemma to exponential function.

Insights into the structure of Hermite-Padé approximations for the exponential.

Abstract

Pad\'e approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of Siegel's lemma to sharpen the estimates of Pad\'e-type approximations, or by finding completely explicit expressions for the yet unknown 'twin type' Hermite-Pad\'e approximations. The appropriate homogeneous matrix equation representing both methods has an $M \times (L+1)$ coefficient matrix, where $M \le L$. The homogeneous solution vectors of this matrix equation give candidates for the Pad\'e polynomials. Due to the Bombieri-Vaaler version of Siegel's lemma, the upper bound of the minimal non-zero solution of the matrix equation can be improved by finding the gcd of all the $M \times M$ minors of the coefficient matrix. In this paper we consider the exponential function and prove that there indeed exists a big common factor of the $M \times M$ minors, giving a possibility to apply the Bombieri-Vaaler version of Siegel's lemma. Further, in the case $M=L$, the existence of this common factor is a step towards understanding the nature of the 'twin type' Hermite-Pad\'e approximations to the exponential function.