Hermite-Padé 逼近与可积性
Hermite-Padé approximation and integrability
AI总结：
本文揭示 Hermite-Padé I 型逼近与 Hirota 离散 KP 系统的内在联系，并借助 Desargues 映射构造解、推广 Wynn 递推和离散时间 Toda 链方程。
摘要：
我们证明，Hermite-Padé I 型逼近问题的解会自然导出 Hirota（离散 Kadomtsev-Petviashvili）系统及其伴随线性问题的一个解子类。我们的结果解释了可积系统理论中的各种组成部分为何会出现在多重正交多项式、数值算法、随机矩阵，以及数学物理和应用数学中其他与 Hermite-Padé 逼近问题相关的分支里。我们还提出了一种基于 Desargues 映射概念的几何算法，用于在有理函数域上的射影空间中构造该问题的解。作为副产品，我们得到了 Wynn 递推的相应推广。我们分离出可给出 Hermite-Padé 问题解的 Hirota 系统边界数据，并表明相应约化降低了该系统的维数。特别地，除 Paszkowski 给出的已知方程外，我们还得到了某些可被视为 Frobenius 恒等式直接类比的方程。我们研究了约化系统在可积性理论中的位置，由此发现了离散时间 Toda 链方程在多维（按变量数目理解）意义下的推广。
英文摘要：
We show that solution to the Hermite-Padé 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-Padé 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-Padé 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.
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