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

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 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.