  Pad\'e 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 Pad\'e-type
approximations, or by finding completely explicit expressions for the yet
unknown 'twin type' Hermite-Pad\'e 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 Pad\'e 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-Pad\'e
approximations to the exponential function.
