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
|Author:||Matala-aho, Tapani1; Seppälä, Louna1|
1Matematiikka, Oulun yliopisto
|Online Access:||PDF Full Text (PDF, 5.2 MB)|
|Persistent link:|| http://urn.fi/urn:nbn:fi-fe2018121851271
|Publish Date:|| 2020-04-17
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 M ≤ L. 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.
Journal of number theory
|Pages:||345 - 383|
|Type of Publication:||
A1 Journal article – refereed
|Field of Science:||
The work of Louna Seppälä was supported by the Magnus Ehrnrooth Foundation.
© 2018 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.