Tapani Matala-aho, Louna Seppälä, Hermite–Thue equation: Padé approximations and Siegel's lemma, Journal of Number Theory, Volume 191, 2018, Pages 345-383, ISSN 0022-314X, https://doi.org/10.1016/j.jnt.2018.03.014.

### Hermite–Thue equation : Padé approximations and Siegel’s lemma

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Author: Matala-aho, Tapani1; Seppälä, Louna1
Organizations: 1Matematiikka, Oulun yliopisto
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 5.2 MB)
Language: English
Published: Elsevier, 2018
Publish Date: 2020-04-17
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# Abstract

Padé 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 Padé-type approximations, or by finding completely explicit expressions for the yet unknown ‘twin type’ Hermite–Padé approximations. The appropriate homogeneous matrix equation representing both methods has an M × (L + 1) coefficient matrix, where ML. The homogeneous solution vectors of this matrix equation give candidates for the Padé 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 × 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 × 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–Padé approximations to the exponential function.

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Series: Journal of number theory
ISSN: 0022-314X
ISSN-E: 1096-1658
ISSN-L: 0022-314X
Volume: 191
Pages: 345 - 383
DOI: 10.1016/j.jnt.2018.03.014