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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 Y. Nakamura's works who used others integrable systems. Our work proves that the finite discrete KP hierarchy completely parametrizes the space

Few decades ago, an unexpected relation between
the control theory and the integrable sys- tems was revealed. Papers [

A large part of this research activity shows that some nonlinear integrable systems have rich information about the moduli space of certain classes of solutions of linear dynamical systems. In particular, they have relation with spaces of certain classes of rational functions. Also according to the state space realization theory, some rational functions can be associated with a controllable and observable linear dynamical system.

A convenient property of these spaces of rational functions is that they can be considered as varieties, thus a question arises about the study of its moduli space. Maybe, in this context, the fundamental role of the integrable systems is its compatibility with families of controllable and observable linear dynamical systems.

During the last ten years, the subject of integrable
system has been enriched in a remarkable way by its
extensions to the other setting, notably, to
discrete case of the KP hierarchy. These developments have originated in the
mathematical physics. In the new settings, many of the classical tools are
available, for example, we point out one of them
which is basic to this paper, the Gauss-Borel decomposition for the discrete KP
hierarchy, proposed by Felipe and Ongay [

In this paper, we use the well-known theory of the
discrete KP hierarchy studied, for instance, in [

We must observe that an interesting property of the finite discrete KP hierarchy is that it contains the full Kostant-Toda equation.

The goals of this section are as follows.

The first goal is to introduce a natural commuting finite hierarchy of flows. We make three comments of this hierarchy. First, it can be defined by a Lax-type operator (matrix) with respect to the shift matrix and its transpose. Second, the Lax matrix introduced admits in certain cases a dressing matrix, in terms of which the hierarchy can be rewritten. Third, there is a Sato-Wilson matrix “to dress” the shift matrix. We mention that the situation is similar to the Sato theory and his dressing technique (pseudodifferential theory).

The second is to review the integrability in the sense of Frobenius for the hierarchy introduced which turns out very simple in this context: the key point in our method is the so-called Gauss-Borel decomposition. It also verifies that the finite discrete KP hierarchy like any integrable system is always related to some kind of group factorization.

Next, we
describe the corresponding Mulase’s approach associated to the finite discrete
KP hierarchy as it is considered in [

Let

Let

The finite discrete KP hierarchy is the Lax system

Now, let us
assume that for an operator defined as in (

From now, we can only consider solutions of (

If there is a
dressing operator such that

In this point, it is very
important to observe that for any given order

It was shown in [

We say that a
matrix

Let

If

Let

Let us consider
the Gauss-Borel factorization of

For

It is easy to
show that if

Let

At this point, it is convenient to inspect the Lax
operator as function of its dressing operators. Let us see it for

From these simple examples, it follows that the unique

Let

We denote the equivalence class to which belongs

Note
that

The functions

Indeed,

Let

We have that

Having in mind
that

On the other
hand, we have that

From (

Let us show that the initial conditions hold.

Since

It is interesting to note that independently
of the selection of

We restrict
ourselves to consider a solution

Next, let us
define a linear dynamical system of parameter

Note that for

Indeed,

Since

By (

Now we will show the observability of (

By (

Let us consider
the function

Since

We see that

As

We can
characterize the flow of (

The flows on

Let us notice
that (

Let

In such way, we
have the nontrivial flow

The equivalence
relation (

The set of solutions

Now, we will
discuss the correspondence between

We can obtain a
flow on

Conversely, any
solution

The first author was supported in part under CONACYT grant 37558E, and in part under the Cuba National Project “Theory and algorithms for the solution of problems in algebra and geometry” and the second author was partially supported by COLCIENCIAS project 395 and CODI project 9889-E01251.