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arXiv · 1805.00750

Hermite-Thue equation: Padé approximations and Siegel's lemma

Tapani Matala-aho·Louna Seppälä
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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 $M \times (L+1)$ coefficient matrix, where $M \le L$. 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 $M \times M$ 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-Padé approximations to the exponential function.

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Tapani Matala-aho, Louna Seppälä. 2018-05-02. Hermite-Thue equation: Padé approximations and Siegel's lemma. https://doi.org/10.1016/j.jnt.2018.03.014

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