The finite discrete KP hierarchy and the rational functions (Q937030)

Summary: The set of all rational functions with any fixed denominator that simultaneously nullify in the infinite point is parametrized by means of a well-known integrable system: a finite-dimensional version of the discrete KP hierarchy. This type of study was originated in \textit{Y. Nakamura}'s works [SIAM J. Math. Anal. 22, No. 6, 1744--1754 (1991; Zbl 0738.93018)] who used others integrable systems. Our work proves that the finite discrete KP hierarchy completely parametrizes the space \(\text{Rat}_{\Lambda }(n)\) of rational functions of the form \(f(x)=q(x)/z^{n}\), where \(q(x)\) is a polynomial of order \(n - 1\) with nonzero independent coefficent. More exactly, it is proved that there exists a bijection from \(\text{Rat}_{\Lambda }(n)\) to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.