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：我们证明，Hermite-Padé I型逼近问题的解自然导出了Hirota（离散Kadomtsev-Petviashvili）系统及其伴随线性问题的一类子解。该结果解释了可积系统理论中的各种要素在多重正交多项式、数值算法、随机矩阵以及涉及Hermite-Padé逼近问题的其他数学物理与应用数学分支中的应用。我们还基于Desargues映射概念提出了几何算法，用于在有理函数域上的射影空间中构造该问题的解，并由此得到了Wynn递推的相应推广。我们分离了Hirota系统的边界数据，这些数据提供了Hermite-Padé问题的解，表明相应的约化降低了系统的维数。特别地，我们得到了某些方程，这些方程除了Paszkowski已知的结果外，可视为Frobenius恒等式的直接类比。我们研究了约化系统在可积性理论中的位置，从而找到了离散时间Toda链方程的多维（在变量数量意义上）推广。