Hermite-Pad\'{e} approximation and integrability

TL;DR

This paper reveals how Hermite-Padé approximation solutions relate to integrable systems like the Hirota system, providing geometric algorithms and new equations that connect approximation theory with integrability and mathematical physics.

Contribution

It establishes a natural link between Hermite-Padé approximation problems and solutions of integrable systems, introduces geometric construction methods, and generalizes classical recurrence relations.

Findings

Solutions correspond to a subclass of Hirota system solutions.

Introduces a geometric algorithm based on Desargues maps.

Generalizes the Wynn recurrence and identifies new integrability equations.

Abstract

We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'{e} approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Pad\'{e} problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.