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.
翻译:我们的结果表明,对于Hermite-Pad\\\\{{{{{{{}}I 类的近似问题,其解决办法自然导致Hirota(Disargues地图概念)系统及其联合线性问题的亚类解决办法。我们的结果解释了在多种正方位多元体、数字变方体、随机矩阵以及数学物理学和应用数学的其他分支中,Hermite-Pad\{{{e}近端问题具有相关性。我们还根据Desargues地图概念,提出了在合理功能领域之上的投影空间问题解决办法的构建几何测算法。作为副产品,我们获得了Wyn复现的对应概括性。我们孤立了为Hermite-Pad\{{{{{}提供解决办法的Hirota系统的边界数据,表明系统相对下降的方位数。特别是,我们获得了某种方位方程式的方程算,除了Paz的直位变数外,我们所认为的方位变数的方位性理论可以降低Paz的方位。