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. 1 approximations). Diagonal Hermite-Padé approximations of the generalised hypergeometric series are quite well established; see de Bruin [5], Huttner [13], Matala-aho [16], Nesterenko [18], and Nikišin [19] for more details. In this work we shall not pursue further this general Padé problem but focus on the Padé approximations to the exponential function instead.1.2. The twin problem. The problem of finding explicit type II Hermite-Padé approximations to the exponential series in the case where the degrees of the polynomials are free parameters and the orders of the remainders are equal was, as mentioned, resolved already by Hermite. But the twin problem (as we call it) of finding explicit type II Hermite-Padé approximations in the case where the degrees of the polynomials are the same but the orders of the remainders are free parameters is still open.Let now l 1 , . . . , l m be positive integers and let α 1 , . . . , α m be distinct variables. Denote α = (α 1 , . . . , α m ) T , l = (l 1 , . . . , l m ) T , and L := l 1 + . . . + l m . Then we may state the twin problem as follows: Find an explicit polynomial B l,0 (t, α), polynomials B l,j (t, α) and remainders S l,j (t, α), j = 1, . . . , m, satisfying the equations