On Some Algebraic Properties of Hermite--Padé Polynomials

Abstract

Let $[f_0,\dots ,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty )\ne 0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Padé polynomials to the given tuple of degrees $\le n$ and $\le mn$ respectively and the corresponding $(m+1)$-multi-indexes with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times (m+1)$ polynomial matrices, $M_1(z),M_2(z)\in \mathrm{GL}(m+1,\mathbb{C}[z])$, generated by type I and type II Hermite--Padé polynomials respectively. Then we have $M_1(z)M_2(z)\equiv I_{m+1}$, where $I_{m+1}$ is the identity $(m+1)\times (m+1)$-matrix. The result is motivated by some novel applications of Hermite--Padé polynomials to the investigation of monodromy properties of Fuchsian systems of differential equations.

4 pages