Hermite–Padé approximation and integrability

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