The Finite Discrete KP Hierarchy and the Rational Functions

Abstract

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 Rat_Λ(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 Rat_Λ(𝑛) to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.

1. Introduction

Few decades ago, an unexpected relation between the control theory and the integrable sys- tems was revealed. Papers [1, 2] which deal with problems related to those discussed here are examples of these researches.

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 [3] as an extension of the Mulase’s algebraic geometric approach to the KP hierarchy [4]. This decomposition allows us to consider on an almost equal footing the cases of semi-infinite and bi-infinite matrices.

In this paper, we use the well-known theory of the discrete KP hierarchy studied, for instance, in [3], restricted to finite matrices to characterize Rat_Λ(𝑛). Thus we will give another example of nonlinear integrable system that also has the property of completely parameterizing some kind of rational function space.

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

2. An Algebraic Geometric Approach for the Finite Discrete KP Hierarchy

The goals of this section are as follows.

(1)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).(2)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 [3]. We omit the proofs because in [3] these were given in a more general background.

Let Λ be the 𝑛 by 𝑛-matrix with ones (the matrix entry equals to 1) in the first upper diagonal and zero in the remaining entriesΛ=01⋯⋯⋯00010⋯0⋮⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱000⋯⋯0100⋯⋯00 (2.1)and Λ^𝑇 its transpose. The matrix Λ is like a shift operator of coordinates for vectors in 𝑅^𝑛. The Λ^𝑘 matrix, 2≤𝑘≤𝑛−1, is a zero matrix except in the 𝑘th upper diagonal where it has ones. Note that Λ^𝑛=Λ^𝑇𝑛=𝑂, where 𝑂 is the zero matrix.

Let 𝐿 be a matrix𝐿=Λ+𝐷₀+_{𝑖=1,…,𝑛−1}𝐷_𝑖(Λ^𝑇)^𝑖, (2.2)where 𝐷_𝑖 are diagonal matrices. The entries of 𝐿 are assumed to be functions depending on parameters 𝑡=(𝑡₁,…,𝑡_𝑛−1).

Definition 2.1. The finite discrete KP hierarchy is the Lax system𝜕𝐿𝜕𝑡_𝑘=[𝐿^𝑘_≥,𝐿],𝑘=1,…,𝑛−1, (2.3)where 𝐿_≥(𝐿_>) denotes the (strictly) upper triangular part of 𝐿, analogously 𝐿_≤(𝐿_<) denotes the (strictly) lower triangular part of 𝐿.

Now, let us assume that for an operator defined as in (2.2), one can find an invertible matrix 𝑆:𝑆=𝐼+𝑆₁Λ^𝑇+𝑆₂(Λ^𝑇)²+⋯+𝑆_𝑛−1(Λ^𝑇)^𝑛−1 (2.4)such that𝐿=𝑆Λ𝑆⁻¹𝑆₋₁,=𝐼+𝑊₁Λ^𝑇+𝑊₂(Λ^𝑇)²+⋯+𝑊_𝑛−1(Λ^𝑇)^𝑛−1 (2.5)(2.6)where in (2.4) and (2.6), 𝑆_𝑖 and 𝑊_𝑖 are diagonal matrices, then 𝐿 will be of Lax type.

From now, we can only consider solutions of (2.3) which are of the Lax type. The operator 𝑆 is called a dressing operator, and the decomposition (2.5) is unique, up to right multiplication by an invertible matrix, taking the form of (2.4) that commutes with Λ. Note that 𝐿=Λ is the most simple solution of (2.3) and for it 𝑆=𝐼.

If there is a dressing operator such that𝜕𝑆𝜕𝑡_𝑘=−𝐿^𝑘_<𝑆,𝑘=1,…,𝑛−1, (2.7)where 𝐿=𝑆Λ𝑆⁻¹, then 𝐿 is of the Lax type, and moreover it satisfies (2.3). Conversely, if 𝐿 is a solution of (2.3) which is of the Lax type, then there exists a dressing operator 𝑆 of 𝐿, such that 𝑆 is solution of (2.7). This operator 𝑆 is called the Sato-Wilson matrix (see [3] for more details).

In this point, it is very important to observe that for any given order 𝑛, it is always possible to find matrices 𝐿 of the form (2.2) which are not of Lax type. It is explained bellow with more details.

It was shown in [3] that from (2.3) the Zakharov-Shabat equations follow 𝜕𝐿^𝑖_≥𝜕𝑡_𝑗−𝜕𝐿^𝑗_≥𝜕𝑡_𝑖=[𝐿^𝑖_≥,𝐿^𝑗_≥],𝑖≠𝑗,𝑖,𝑗=1,…,𝑛−1. (2.8)

We say that a matrix 𝑀 admits a Gauss-Borel decomposition if 𝑀 can be written in the form𝑀=𝑀_≤𝑀_≥, (2.9)where 𝑀_≤ has ones on the principal diagonal and 𝑀_≥ has nonzero elements on the principal diagonal. Decomposition (2.9) is equivalent to𝑀=(𝐼+𝐺_<𝐺₀⁻¹)(𝐼+𝐺₀+𝐺_>), (2.10)where 𝐺₀ is a diagonal matrix with nonzero elements. It can be proved that the Gauss-Borel decomposition is unique. 𝑀 admits the Gauss-Borel decomposition if and only if 𝑀_𝑘≠0,𝑘=1,…,𝑛, where 𝑀_𝑘 is the determinant of the 𝑘-order principal submatrix. Of particular interest will be the matrix space 𝑀^∗ of matrices 𝑀 depending on 𝑡, admitting a Gauss-Borel decomposition and for which 𝑀_≥(0)=𝐼, where 𝐼 is the identity matrix.

Let 𝑍 be the formal 1-form of 𝐿 given by𝑍=^𝑛−1_𝑘=1𝐿^𝑘_≥𝑑𝑡_𝑘. (2.11)

If 𝐿 satisfies (2.3), then 𝑍 satisfies1𝑑𝑍=2[𝑍,𝑍] (2.12)which is equivalent to the Zakharov-Shabat equations, where by definition[𝑍,𝑍]=^𝑛−1_𝑖,𝑗=1[𝐿^𝑖_≥,𝐿^𝑗_≥]𝑑𝑡_𝑖𝑑𝑡_𝑗. (2.13)

LetΩ=^𝑛−1_𝑘=1Λ^𝑘𝑑𝑡_𝑘 (2.14)which is a trivial solution of (2.12). Each solution of (2.3) yields a solution of the linear equation𝑑𝑈=Ω𝑈 (2.15)in 𝑀^∗ and conversely for a solution of (2.15) in 𝑀^∗ we can build a matrix 𝐿 which will be a solution of (2.3). The solutions of (2.15) take the form𝑈=𝑒^{∑𝑛−1𝑘=1Λ𝑘𝑡𝑘}𝑈₀. (2.16)

Let us consider the Gauss-Borel factorization of 𝑈:𝑈=𝑆⁻¹𝑌, (2.17) where 𝑆⁻¹ is a lower triangular matrix with ones on the principal diagonal and 𝑌 is an upper triangular matrix with nonzero elements on the principal diagonal.

For 𝑈∈𝑀^∗ we have that𝑈₀=𝑈(0)=𝑆⁻¹(0)𝑌(0)=𝑆⁻¹(0)𝐼=𝑆₀⁻¹, (2.18)then𝑈=𝑒^{∑𝑛−1𝑘=1Λ𝑘𝑡𝑘}𝑈₀=𝑒^{∑𝑛−1𝑘=1Λ𝑘𝑡𝑘}𝑆₀⁻¹, (2.19)where 𝑆₀⁻¹ is a matrix, which takes the form of (2.4).

It is easy to show that if 𝑈∈𝑀^∗, then in (2.17) 𝑆 is a Sato-Wilson operator, therefore using the Gauss-Borel factorization of 𝑈 and doing 𝐿=𝑆Λ𝑆⁻¹, we obtain a solution of (2.3) for which 𝐿(0)=𝑆₀Λ𝑆₀⁻¹=𝐿₀, (2.20)(see [3]).

3. Rational Functions Induced by the Finite Discrete KP Hierarchy Solutions

Let rat_𝑝(𝑛) be the space of rational functions of grade 𝑛 and fixed denominator 𝑝(𝑧)=𝑧^𝑛+𝑝_𝑛−1𝑧^𝑛−1+⋯+𝑝₀. It is possible to see that an element 𝑓(𝑧)∈rat_𝑝(𝑛) always admits a unique factorization𝑓(𝑧)=𝑞(𝑧)𝑝(𝑧)=𝐶^𝑇₀(𝑧𝐼_𝑛−𝐿_𝑝)⁻¹𝐵₀, (3.1)where 𝐶^𝑇₀=(1,0,…),𝐵^𝑇₀=(𝑏_𝑛−1⋅𝑏_𝑛−2,…,𝑏₀),𝑞(𝑧)=𝑏_𝑛−1𝑧^𝑛−1+𝑏_𝑛−2𝑧^𝑛−2+⋯+𝑏₀, and𝐿_𝑝=−𝑝_𝑛−110⋯0−𝑝_𝑛−20⋱⋱⋮⋮⋮⋱⋱0−𝑝₁0⋯⋱1−𝑝₀0⋯⋯0, (3.2)this matrix 𝐿_𝑝 will be denominated by 𝑝-operator (note that 𝐿_𝑝 is of the form (2.2)). Observe that the characteristic polynomials of 𝐿_𝑝 is equal to the given 𝑝(𝑧).

At this point, it is convenient to inspect the Lax operator as function of its dressing operators. Let us see it for 𝑛=2 and 𝑛=3: 𝑆𝑛=2𝑛=3₋₁𝑆𝐿10𝑎1100𝑎10𝑐𝑏110−𝑎1100−𝑎10𝑎𝑏−𝑐−𝑏1−𝑎1−𝑎²𝑎𝑎10−𝑎²+𝑐−𝑎+𝑏1(𝑎²−𝑐)𝑏−𝑎𝑐(𝑎𝑏−𝑐)−𝑏².−𝑏 (3.3)

From these simple examples, it follows that the unique 𝑝-operator that is also a Lax operator is justly 𝐿_𝑝 = Λ. This assertion holds also for any arbitrary dimension of matrix 𝐿_𝑝 and it shows that rat(𝑛) (its definition is given next) is the unique space that could be characterized by this hierarchy, using the Gauss-Borel factorization.

Let rat(𝑛) be a space of rational functions 𝑓(𝑧)∈rat_𝑝(𝑛) such that𝑎𝑓(𝑧)=¹𝑧^𝑛−1+𝑎₂𝑧^𝑛−2+⋯+𝑎_𝑛𝑧^𝑛=𝑞(𝑧)𝑧^𝑛 (3.4)with 𝑎_𝑛≠0 and 𝑎_𝑖∈ℝ for any 𝑖. On this space, it is usual to introduce the equivalence relation𝑓∼𝛼𝑓𝛼𝜖𝑅⧵{0}. (3.5)

We denote the equivalence class to which belongs 𝑓(𝑧) and by [𝑓] and the set rat(𝑛)⧵∼={[𝑓]∶𝑓𝜖rat(𝑛)} by Rat_Λ(𝑛). As mentioned before, any element of rat(𝑛) can be written in the form𝑓(𝑧)=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝐵₀, (3.6)where 𝐶^𝑇₀=(1,0,…,0),𝐵^𝑇₀=(𝑎₁,𝑎₂,…,𝑎_𝑛) are constant vectors in 𝑅^𝑛, and 𝑎_𝑖 are the coef- ficients of 𝑓(𝑧).

Note that𝛼𝑓(𝑧)=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹(𝛼𝐵₀). (3.7)

The functions 𝑓𝜖rat(𝑛) expressed according to (3.6) can be considered as the transfer functions of the linear dynamical system𝑑𝑥𝑑𝜏=Λ𝑥(𝜏)+𝐵₀𝑢(𝜏),𝑦(𝜏)=𝐶^𝑇₀𝑥(𝜏). (3.8)

Proposition 3.1. The linear dynamical system (3.8) associated to 𝑓(𝑧)=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝐵₀𝜖rat(𝑛) is controllable and observable.

Proof. Indeed,rang((𝐵₀)(Λ𝐵₀)⋯(Λ^𝑛−1𝐵₀))=𝑛 (3.9)because𝑎₁𝑎₂𝑎_𝑛𝑎₂𝑎₃0𝑎.....⋯...._𝑛−1𝑎_𝑛0𝑎_𝑛00=−(𝑎^𝑛_𝑛)≠0 (3.10)if 𝑎_𝑛≠0. This fact implicates the controllability of (3.8). Alsorang((𝐶₀)(Λ^𝑇𝐶₀)⋯(Λ^𝑇𝑛−1𝐶₀))=𝑛 (3.11)because (Λ^𝑇)^𝑖𝐶₀,𝑖=0,…,𝑛−1, is the canonical base on 𝑅^𝑛. This fact implies that (3.8) is observable.

Proposition 3.2. Let 𝑈∈𝑀^∗, 𝑈=𝑆⁻¹𝑌 and 𝐵, 𝐶 be defined as follows:𝐵(𝑡₁,…,𝑡_𝑛−1)=𝑌(𝑡₁,…,𝑡_𝑛−1)𝐵₀,𝐶(𝑡₁,…,𝑡_𝑛−1)=(𝑆⁻¹(𝑡₁,…,𝑡_𝑛−1))^𝑇𝐶₀, (3.12)then 𝐵 and 𝐶 satisfy the linear equations 𝜕𝐵𝜕𝑡_𝑘=𝐿^𝑘_≥𝐵,𝜕𝐶𝜕𝑡_𝑘=(𝐿^𝑘_<)^𝑇𝐶,𝑘=1,…,𝑛−1,𝐵(0)=𝐵₀,𝐶(0)=𝐶₀, (3.13)where 𝐿=𝑆Λ𝑆⁻¹.

Proof. We have that𝜕𝐵𝜕𝑡_𝑘=𝜕𝑌𝜕𝑡_𝑘𝐵₀ (3.14)for 𝑘=1,…,𝑛−1. Being 𝑈=𝑆⁻¹𝑌, we obtain𝜕𝑈𝜕𝑡_𝑘=𝜕𝑆⁻¹𝜕𝑡_𝑘𝑌+𝑆⁻¹𝜕𝑌𝜕𝑡_𝑘 (3.15)and since𝑆⁻¹𝑆=𝐼𝜕𝑆⁻¹𝜕𝑡_𝑘𝑆+𝑆⁻¹𝜕𝑆𝜕𝑡_𝑘=0,𝜕𝑈𝜕𝑡_𝑘=−𝑆⁻¹𝜕𝑆𝜕𝑡_𝑘𝑆⁻¹𝑌+𝑆⁻¹𝜕𝑌𝜕𝑡_𝑘. (3.16)
Having in mind that 𝑆 is a Sato-Wilson operator and 𝜕𝑈/𝜕𝑡_𝑘=Λ^𝑘𝑈,Λ^𝑘𝑈=𝜕𝑈𝜕𝑡_𝑘=−𝑆⁻¹𝜕𝑆𝜕𝑡_𝑘𝑆⁻¹𝑌+𝑆⁻¹𝜕𝑌𝜕𝑡_𝑘,𝑆Λ^𝑘𝑈+𝜕𝑆𝜕𝑡_𝑘𝑆⁻¹𝑌=𝜕𝑌𝜕𝑡_𝑘,(𝑆Λ^𝑘𝑆⁻¹)𝑌−𝐿^𝑘_<𝑌=𝜕𝑌𝜕𝑡_𝑘,𝐿^𝑘_≥𝑌=(𝐿^𝑘−𝐿^𝑘_<)𝑌=𝜕𝑌𝜕𝑡_𝑘, (3.17)then𝜕𝐵𝜕𝑡_𝑘=𝜕𝑌𝜕𝑡_𝑘𝐵₀=𝐿^𝑘_≥𝑌𝐵₀=𝐿^𝑘_≥𝐵. (3.18)
On the other hand, we have that𝜕𝐶𝜕𝑡_𝑘=𝜕(𝑆⁻¹)^𝑇𝜕𝑡_𝑘𝐵₀,𝐶=(𝑆⁻¹)^𝑇𝐶₀=(𝑆^𝑇)⁻¹𝐶₀ (3.19)for 𝑘=1,…,𝑛−1. Since 𝑆 verifies (2.7), it is Sato-Wilson matrix then𝜕𝑆^𝑇𝜕𝑡_𝑘=−𝑆^𝑇𝐿^𝑘_<. (3.20)
From (3.19) and (3.20), its follows that𝜕𝐶𝜕𝑡_𝑘=𝜕(𝑆⁻¹)^𝑇𝜕𝑡_𝑘𝐶₀=𝜕(𝑆^𝑇)⁻¹𝜕𝑡_𝑘𝐶₀=(𝐿^𝑘_<)^𝑇(𝑆^𝑇)⁻¹𝐶₀=(𝐿^𝑘_<)^𝑇𝐶. (3.21)
Let us show that the initial conditions hold.
Since 𝐵=𝑌𝐵₀ and 𝑌(0)=𝐼, then 𝐵(0)=𝐵₀. On other hand, due to 𝐶=(𝑆⁻¹)^𝑇𝐶₀, we have that 𝐶(0)=(𝑆⁻¹(0))^𝑇𝐶₀=(𝑆₀⁻¹)^𝑇𝐶₀. As (𝑆₀⁻¹)^𝑇 is an upper triangular matrix with ones on the diagonal, then (𝑆₀⁻¹)^𝑇𝐶₀=𝐶₀.

Remark 3.3. It is interesting to note that independently of the selection of 𝑆₀⁻¹ as initial condition in the factorization of 𝑈, the flow for 𝐶 always begins in 𝐶₀.

We restrict ourselves to consider a solution 𝐿 of (2.3) to which 𝐿_𝐼(0)=Λ and we denote it by 𝐿_𝐼. In this case, we must take 𝑆₀=𝐼 by (2.20).

Next, let us define a linear dynamical system of parameter 𝑡=(𝑡₁,…,𝑡_𝑛−1) using 𝐵(𝑡), 𝐶(𝑡), and 𝐿_𝐼(𝑡) in the following way𝑑𝑑𝜏𝑥(𝜏,𝑡)=𝐿(𝜏,𝑡)𝑥(𝜏,𝑡)+𝐵(𝜏,𝑡)𝑢(𝜏,𝑡),𝑦(𝜏,𝑡₁)=𝐶^𝑇(𝜏,𝑡)𝑥(𝜏,𝑡). (3.22)

Note that for 𝑡₁=⋯=𝑡_𝑛−1=0, we obtain (3.8).

Proposition 3.4. The linear dynamical system (3.22) is controllable and observable.

Proof. Indeed,rang((𝐵)(𝐿𝐵)⋯(𝐿^𝑛−1𝐵))=rang((𝑌𝐵₀)(𝑆Λ𝑆⁻¹𝑌𝐵₀)⋯(𝑆Λ^𝑛−1𝑆⁻¹𝑌𝐵₀))=rang(𝑆((𝑈𝐵₀)(Λ𝑈𝐵₀)⋯(Λ^𝑛−1𝑈𝐵₀))). (3.23)
Since Λ and 𝑈 commute, thenrang((𝐵)(𝐿𝐵)⋯(𝐿^𝑛−1𝐵))=rang(𝑆𝑈(𝐵₀)(Λ𝐵₀)⋯(Λ^𝑛−1𝐵₀))=rang(𝑌(𝐵₀)(Λ𝐵₀)⋯(Λ^𝑛−1𝐵₀)). (3.24)
By (3.9) and due to det𝑌≠0, we have thatrang(𝑌(𝐵₀)(Λ𝐵₀)⋯(Λ^𝑛−1𝐵₀))=𝑛 (3.25)from whererang((𝐵)(𝐿𝐵)⋯(𝐿^𝑛−1𝐵))=𝑛 (3.26)then (3.22) is controllable.
Now we will show the observability of (3.32).rang((𝐶)(𝐿𝐶)⋯(𝐿^𝑛−1𝐶))=rang(((𝑆^𝑇)⁻¹𝐶₀)((𝑆^𝑇)⁻¹Λ^𝑇𝑆^𝑇(𝑆^𝑇)⁻¹𝐶₀)⋯((𝑆^𝑇)⁻¹(Λ^𝑇)^𝑛−1𝑆^𝑇(𝑆^𝑇)⁻¹𝐶₀))=rang((𝑆^𝑇)⁻¹(𝐶₀(Λ^𝑇𝐶₀)⋯((Λ^𝑇)^𝑛−1𝐶₀)). (3.27)
By (3.11) and since det(𝑆^𝑇)⁻¹≠0, we haverang((𝑆^𝑇)⁻¹(𝐶₀(Λ^𝑇𝐶₀)⋯((Λ^𝑇)^𝑛−1𝐶₀))=𝑛 (3.28)from whererang((𝐶)(𝐿𝐶)⋯(𝐿^𝑛−1𝐶))=𝑛. (3.29)

Let us consider the function𝐹(𝑧,𝑡₁,…,𝑡_𝑛−1)=𝐶^𝑇(𝑧𝐼−𝐿_𝐼)⁻¹𝐵=𝐶^𝑇(𝑆(𝑧𝐼−Λ)𝑆⁻¹)⁻¹𝐵=𝐶^𝑇𝑆(𝑧𝐼−Λ)⁻¹𝑆⁻¹𝐵=((𝑆⁻¹)^𝑇𝐶₀)^𝑇𝑆(𝑧𝐼−Λ)⁻¹𝑆⁻¹𝐵=𝐶^𝑇₀𝑆⁻¹𝑆(𝑧𝐼−Λ)⁻¹𝑆⁻¹𝑌𝐵₀=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝑈_𝐼𝐵₀. (3.30)

Since 𝑈_𝐼(0)=𝐼, then𝐹(𝑧,0)=𝑓(𝑧)=𝐶^𝑇₀(𝑧𝐼−𝐿_𝐼(0))⁻¹𝐵₀. (3.31)

We see that 𝐹(𝑧,𝑡₁,…,𝑡_𝑛−1)𝜖rat(𝑛). From (3.30), we have that𝐹(𝑧,𝑡₁,…,𝑡_𝑛−1𝑏)=¹𝑧^𝑛−1+𝑏₂𝑧^𝑛−2+⋯+𝑏_𝑛𝑧^𝑛=𝑞_𝑛−1(𝑧,𝑡₁,…,𝑡_𝑛−1)𝑧^𝑛. (3.32)

As 𝑈_𝐼 is an upper triangular matrix with ones on the diagonal, then the independent terms of 𝑞_𝑛−1 and of the numerator of 𝑓(𝑧) coincide.

We can characterize the flow of (3.30). Taking derivate respect to 𝑡_𝑘, we obtain𝜕𝐹𝜕𝑡_𝑘=𝐶^𝑇₀(𝑧𝐼−𝐿_𝐼)⁻¹𝜕𝑈_𝐼𝜕𝑡_𝑘𝐵₀,𝑘=1,…,𝑛−1, (3.33)doing 𝑉=𝑈_𝐼𝐵₀, we have𝐹=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝑉. (3.34)

The flows on rat(𝑛) determined by 𝐿_𝐼 have the form𝜕𝑉𝜕𝑡_𝑘=Λ^𝑘𝑉,𝑘=1,…,𝑛−1, (3.35)where 𝑉 defines the numerator of 𝐹 and it has the form𝑉=𝑒^{∑𝑛−1𝑘=1Λ𝑘𝑡𝑘}𝐵₀. (3.36)

Let us notice that (3.35) is similar to the flows considered by Brockett and Faybusovich in [1].

Let (𝐿_𝐼(𝑡),𝐵(𝑡),𝐶(𝑡)) be a triple, where 𝐿_𝐼(𝑡) is the above fixed solution of (2.3); 𝐵(𝑡) and 𝐶(𝑡) are solutions of (3.13). This triple gives a flow of (3.30) on rat(𝑛) defined by the initial conditions 𝐿_𝐼(0)=Λ,𝐵(0)=𝐵₀,𝐶(0)=𝐶₀ that determine the triple (Λ,𝐵₀,𝐶₀).

In such way, we have the nontrivial flow (Λ,𝐵₀,𝐶₀)→(𝐿_𝐼,𝐵,𝐶) on 𝑅^2𝑛+𝑛2.

The equivalence relation (3.5) induces an equivalence relation ≺ on the set of triples (Λ,𝐵₀,𝐶₀) such that(Λ,𝐵¹₀,𝐶₀)≺(Λ,𝐵²₀,𝐶₀),if𝐵¹₀=𝛼𝐵²₀,𝛼𝜖𝑅⧵{0}. (3.37)

Definition 3.5. The set of solutions (𝐿_𝐼(𝑡),𝐵(𝑡),𝐶(𝑡)) of compatible systems (2.3) and (3.13) for any equivalence class [(Λ,𝐵₀,𝐶₀)] that satisfies (3.6) is denominated by the moduli space of systems (2.3) and (3.13). We denote this moduli space as 𝑀_Λ.

Now, we will discuss the correspondence between Rat_Λ(𝑛) and 𝑀_Λ and we will built a one-to-one mapping from Rat_Λ(𝑛) to 𝑀_Λ.

We can obtain a flow on Rat_Λ(𝑛) by means of (3.35), determined by 𝑓(𝑧) in the following way:𝐹(𝑧,𝑡₁,…,𝑡_𝑛−1)=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝑉=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝑒^{∑𝑛−1𝑘=1Λ𝑘𝑡𝑘}𝐵₀=𝐶^𝑇₀(𝑧𝐼−Λ)⁻¹𝑆⁻¹𝑌𝐵₀=((𝑆⁻¹)^𝑇𝐶₀)^𝑇(𝑧𝐼−𝐿_𝐼)⁻¹𝑌𝐵₀=𝐶^𝑇(𝑧𝐼−𝐿_𝐼)⁻¹𝐵, (3.38)where 𝐶 and 𝐵 satisfy (3.13), 𝐿_𝐼 satisfy (2.3), and 𝐿_𝐼(0)=Λ. Thus any [𝑓]𝜖Rat_Λ(𝑛) defines a solution (𝐿_𝐼,𝐵,𝐶) determined by the initial conditions [(Λ,𝐵₀,𝐶₀)]. So we have a mapping𝛽∶Rat_Λ(𝑛)→𝑀_Λ. (3.39)

Conversely, any solution (𝐿_𝐼,𝐵,𝐶)∈𝑀_Λ induces a flow (3.30) on Rat_Λ(𝑛) which originates into 𝑓(𝑧) at 𝑡=0. Furthermore, by (3.31) we obtain the class [(Λ,𝐵₀,𝐶₀)]. Thus 𝛽 is an onto mapping form Rat_Λ(𝑛) to 𝑀_Λ.

Theorem 3.6. The mapping 𝛽 defined in (3.39) is a bijection.

Acknowledgments

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.