Integer solutions of integral inequalities and H-invariant Jacobian Poisson structures

We study the Jacobian Poisson structures in any dimension invariant with respect to the discrete Heisenberg group. The classification problem is related to the discrete volume of suitable solids. Particular attention is given to dimension 3 whose simplest example is the Artin-Schelter-Tate Poisson tensors respectively.


Introduction
This paper continues the author's program of studies the Heisenberg invariance properties of polynomial Poisson algebras which were started in [18] and extended in [21,22]. Formally speaking, we consider the polynomials in n variables C[x 0 , x 1 , · · · , x n−1 ] over C and the action of some subgroup H n of GL n (C) generated by the shifts operators x i −→ x i+1 (modZ n ) and by the operators x −→ ε i x i , where ε n = 1. We are interested in the polynomial Poisson brackets on C[x 0 , x 1 , · · · , x n−1 ] which are "stable" under this actions ( we will give more precise definition below). The most famous examples of the Heisenberg invariant polynomial Poisson structures are the Sklyanin-Odesskii-Feigin -Artin-Tate quadratic Poisson brackets known also as the elliptic Poisson structures. One can also think about these algebras like the"quasi-classical limits" of elliptic Sklyanin associative algebras. These is a class of Noetherian graded associative algebras which are Koszul, Cohen-Macaulay and have the same Hilbert function as a polynomial ring with n variables. The above mentioned Heisenberg group action provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines an H n -action on the elliptic quadratic Poisson structures on P n . The latter are identified with Poisson structures on some moduli spaces of the degree n and rank k + 1 vector bundles with parabolic structure (= the flag 0 ⊂ F ⊂ C k+1 on the elliptic curve E). We will denote this elliptic Poisson algebras by q n;k (E). The algebras q n;k (E) arise in the Feigin-Odesskii "deformational" approach and form a subclass of polynomial Poisson structures. A comprehensive review of elliptic algebras can be found in ( [20]) to which we refer for all additional information. We will mention only that as we have proved in [22] all elliptic Poisson algebras (being in particular Heisenberg-invariant) are unimodular.
Another interesting class of polynomial Poisson structures consists of so-called Jacobian Poisson structures (JPS). These structures are a special case of Nambu-Poisson structures. Their rank is two and the Jacobian Poisson bracket {P, Q} of two polynomials P and Q is given by the determinant of Jacobi matrix of functions (P, Q, P 1 , ..., P n−2 ). The polynomials P i , 1 ≤ i ≤ n − 2 are Casimirs of the bracket and under some mild condition of independence are generators of the centrum for the Jacobian Poisson algebra structure on C[x 0 , . . . , x n−1 ]. This type of Poisson algebras was intensively studied (due to their natural origin and relative simplicity) in a huge number of publications among which we should mention [28], [16] [13], [14], [29] and [18].
There are some beautiful intersections between two described types of polynomial Poisson structures: when we are restricting ourselves to the class of quadratic Poisson brackets then there are only Artin-Schelter-Tate (n = 3) and Sklyanin (n = 4) algebras which are both elliptic and Jacobian. It is no longer true for n > 4. The relations between the Sklyanin Poisson algebras q n,k (E) whose centrum has dimension 1 (for n odd) and 2 (for n even) in the case k = 1 and is generated by l = gcd(n, k + 1) Casimirs for q n,k (E) for k > 1 are in general quite obscure. We can easily found that sometimes the JPS structures correspond to some degenerations of the Sklyanin elliptic algebras. One example of such JPS for n = 5 was remarked in [14] and was attributed to so-called Briesckorn-Pham polynomials for n = 5: It is easy to check that the homogeneous quintic P = P 1 P 2 P 3 (see sect. 3.2) defines a Casimir for some "rational degeneration of (one of) elliptic algebras q 5,1 (E) and q 5,2 (E) if it satisfies the H−invariance condition. In this paper we will study the Jacobian Poisson structures in any number of variables which are Heisenberg-invariant and we relate all such structures to some graded sub-vector space H of polynomial algebra. This vector space is completely determined by some enumerative problem of a number-theoretic type. More precisely, the homogeneous subspace H i of H of degree i is in bijection with integer solutions of a system of Diophant inequalities. Geometric interpretation of the dimension of H i is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3, H is a subalgebra of polynomial algebra with 3 variables and all JPS are given by this space. We solve explicitly the enumerative problem in this case and obtain a complete classification of the H-invariant not necessarily quadratic Jacobian Poisson algebras with three generators. As a by product we explicitly compute the Poincaré series of H. In this dimension we observe that the H-invariant JPS of degree 5 is given by the Casimir sextic a, b, c, d ∈ C. This structure is a "projectively dual" to the Artin-Schelter-Tate elliptic Poisson structure which is the H-invariant JPS given by the cubic where γ ∈ C. In fact the algebraic variety E ∨ : {P ∨ = 0} ∈ P 2 is the (generically) projectively dual to the elliptic curve E : {P = 0} ⊂ P 2 . The paper is organized as follows: in Section 1 we remind a definition of the Heisenberg group in the Schroedinger representation and describe its action on Poisson polynomial tensors and also the definition of JPS. In Section 2 we treat the above mentioned enumerative problem in dimension 3. The last section concerns the case of any dimension. Here we discuss some possible approaches to the general enumerative question.

Preliminary facts
Throughout of this paper, K is a field of characteristic zero. Let us start by remaining some elementary notions of the Poisson geometry.

Poisson algebras and Poisson manifold
Let R be a commutative K-algebra. One says that R is a Poisson algebra if R is endowed with a Lie bracket, indicated with {·, ·}, which is also a biderivation. One can also say that R is endowed with a Poisson structure and therefore the bracket {·, ·} is called the Poisson bracket. Elements of the center are called Casimirs: a ∈ R is a Casimir if {a, b} = 0 for all b ∈ R.
Poisson structure π is defined by the formula: or more explicitly (mod Z 4 ): Sklyanin had introduced this Poisson algebra which carried today his name in a Hamiltonian approach to the continuous and discrete integrable Landau-Lifshitz models ( [24], [25]). He showed that the Hamiltonian structure of the classical model is completely determined by two quadratic "Casimirs". The Sklyanin Poisson algebra is also called elliptic since its relations with an elliptic curve. The elliptic curve enters in the game from the geometric side. The symplectic foliation of Sklyanin's structure is too complicated. This is because the structure is degenerated and looks quite different from a symplectic one. But the intersection locus of two Casimirs in the affine space of dimension four (one can consider also the projective situation) is an elliptic curve E given by two quadrics q 1,2 . We can think about this curve E as a complete intersection of the couple q 1 = 0, q 2 = 0 embedded in CP 3 (as it was observed in Sklyanin's initial paper). A possible generalization one can be obtained considering n − 2 polynomials Q i in K n with coordinates x i , i = 0, ..., n − 1. We can define a bilinear differential operation: by the formula This operation, which gives a Poisson algebra structure on K[x 1 , ..., x n ], is called a Jacobian Poisson structure (JPS), and it is a partial case of more general n − m-ary Nambu operation given by an antisymmetric n − m-polyvector field introduced by Y.Nambu [16] and was extensively studied by L. Takhtajan [28]. The polynomials Q i , i = 1, ..., n − 2 are Casimir functions for the brackets (1).
There exists a second generalization of the Sklyanin algebra that we will describe briefly in the next sub-section (see for details [20]).

Elliptic
Poisson algebras q n (E, η) and q n,k (E, η) (We report here this subsection from [21] for sake of self-consistency). These algebras, defined by Odesskii and Feigin, arise as quasi-classical limits of elliptic associative algebras Q n (E, η) and Q n,k (E, η) [7,17]. Let Γ = Z + τ Z ⊂ C, be an integral lattice generated by 1 and τ ∈ C, with Imτ > 0. Consider the elliptic curve E = C/Γ and a point η on this curve. In their article [17], given k < n, mutually prime, Odesskii and Feigin construct an algebra, called elliptic, Q n,k (E, η), as an algebra defined by n generators {x i , i ∈ Z/nZ} and the following relations where θ α are theta functions [17]. These family of algebras has the following properties : 1. The center of the algebra Q n,k (E, η), for generic E and η, is the algebra of polynomial of m = pgcd(n, k + 1) variables of degree n/m; 6. the algebras Q n,k (E, η) are deformations of polynomial algebras. The associated Poisson structure is denoted by q n,k (E, η); 7. among the algebras q n,k (E, η), only q 3 (E, η) (the Artin-Schelter-Tate algebra) and the Sklyanin algebra q 4 (E, η) are Jacobian Poisson structures.

The G-invariant Poisson structures
Let G be a group acting on a Poisson algebra R.
In other words, for every g ∈ G the morphism ϕ g : R −→ R, a → g · a is an automorphism and the following diagram is a commutative:

The H-invariant Poisson structures
In their paper [21], the authors introduced the notion of H-invariant Poisson structures. That is a special case of a G-invariant structure when G in the finite Heisenberg Group and R is the polynomial algebra. Let us remind this notion. Let V be a complex vector space of dimension n and e 0 , · · · , e n−1 -a basis of V . Take the n−th primitive root of unity ε = e 2πi n .
For simplicity the H n -invariance condition will be referred from now on just as Hinvariance. An H-invariant Poisson bracket on A = C[x 0 , x 1 , · · · , x n−1 ] is nothing but a bracket on A which satisfy the following: for all i, j ∈ Z/nZ.
The τ invariance is, in some sense, a "discrete" homogeneity.

H-invariant JPS in dimension 3
We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras.
be the polynomial algebra with 3 generators. For every P ∈ A we have a JPS π(P ) on R given by the formula: where (i, j, k) ∈ Z/3Z is a cyclic permutation of (0, 1, 2). Let H the set of all P ∈ A such that π(P ) is an H-invariant Poisson structure.

Proposition 2.2. H is a subalgebra of R.
Proof. Let P, Q ∈ H. It is clear that for all α, β ∈ C, αP + βQ belongs to H. Let us denote by {·, ·} F the JPS associated to the polynomial F ∈ R. It is easy to verify that {x i , x j } P Q = P {x i , x j } Q + Q{x i , x j } P . Therefore, it is clear that the H-invariance condition is verified for the JPS associated to the polynomial P Q.
We endow H with the usual grading of the polynomial algebra R. For F an element of R, we denote by ̟(F ) its usual weight degree. We denote by H i the homogeneous subspace of H of degree i. Proof. First of all H 0 = C. We suppose now i = 0. Let P ∈ H i , P = 0. Then ̟(P ) = i. It follows from Proposition (1.2) and the definition of the Poisson brackets, that there exists s ∈ N such that ̟(x i , x j ) = 2 + 3s. The result follows from the fact that ̟(x i , x j ) = ̟(P ) − 1.
Proof. This is a direct consequence of proposition (2.3).
Proposition 2.5. The system equation (5) has as solutions the following set: where r = 1 + s, s ′ and s ′′ live in the polygon given by the following inequalities in Remark 2.1. For r = 1, one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the Casimir P = α(x 3 0 + x 3 1 + x 3 2 ) + βx 0 x 1 x 2 , α, β ∈ C. Let suppose α = 0, then it can take the form: where γ ∈ C. The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following non-algebraic changes of variables: Let The polynomial P in the coordinates (y 0 ; y 1 ; y 2 ) has the form P = (y 3 0 + y 3 1 y 2 + y 2 2 ) + γy 0 y 1 y 2 .
The Poisson bracket is also polynomial (which is not evident at all!) and has the same form: where (i, j, k) is the cyclic permutation of (0, 1, 2). This JPS structure is no longer satisfied the Heisenberg invariance condition. But it is invariant with respect the following toric action: (C * ) 3 × P 2 → P given by the formula: Put deg y 0 = 2; deg y 2 = 1; deg y 2 = 3. Then the polynomial P is also homogeneous in (y 0 ; y 1 ; y 2 ) and defines an elliptic curve P = 0 in the weighted projective space WP 2;1;3 .
The similar change of variables defines the JPS structure invariant with respect to the torus action (C * ) 3 × P 2 → P given by the formula: and related to the elliptic curve 1/3(z 2 2 + z 2 0 z 2 + z 0 z 3 1 ) + kz 0 z 1 z 2 = 0 in the weighted projective space WP 1;1;2 .
These structures had appeared in [18], their Poisson cohomology were studied by A. Pichereau ([23]) and their relation to the non-commutative del Pezzo surfaces and Calabi-Yau algebras were discussed in [11].
The corresponding Poisson bracket takes the form: where i, j, k are the cyclic permutations of 0, 1, 2.
This new JPS should be considered as the "projectively dual" to the Artin-Schelter-Tate JPS since the algebraic variety E ∨ : P ∨ = 0 is generically the projective dual curve in P 2 to the elliptic curve To establish the exact duality and the explicit values of the coefficients we should use (see [9] ch.1) the Schläfli's formula for the dual of a smooth plane cubic E = 0 ⊂ P 2 . The coordinates (p 0 : p 1 : p 2 ) ∈ P 2 * of a point p ∈ P 2 * satisfies to the sextic relation Set S r = T r ∩ N 2 . S r = S 1 r ∪ S 2 r , S 1 r = {(x, y) ∈ S r : 0 ≤ x ≤ r} and S 1 r = {(x, y) ∈ S r : r < x ≤ 2r}. dimH 3r = card(S 1 r ) + card(S 2 r ). Proposition 2.7.
if r is even if r is odd.
if r is even if r is odd.

H-invariant JPS in any dimension
In order to formulate the problem in any dimension, let us remind some numbertheoretic notions concerning the enumeration of nonnegative integer points in a polytope or more generally discrete volume of a polytope.

Enumeration of integer solutions to linear inequalities
In their papers [2,4], the authors study the problem of nonnegative integer solutions to linear inequalities as well as their relation with the enumeration of integer partitions and compositions. Define the weight of a sequence λ = (λ 0 , λ 2 , · · · , λ n−1 ) of integers to be |λ| = λ 0 +· · ·+ λ n−1 . If sequence λ of weight N has all parts nonnegative, it is called a composition of N ; if, in addition, λ is a non increasing sequence, we call it a partition of N .
Given an r × n integer matrix C = [c i,j ], (i, j) ∈ ({−1} ∪ Z/rZ) × Z/nZ, consider the set S C of nonnegative integer sequences λ = (λ 1 , λ 2 , · · · , λ n ) satisfying the constraints The associated full generating function is defined as follows: This function "encapsulates" the solution set S C : the coefficient of q N in F C (qx 0 , qx 1 ,· · · , qx n−1 ) is a "listing" (as the terms of a polynomial) of all nonnegative integer solutions to (13) of weight N and the number of such solutions is the coefficient of q N in F C (q, q, · · · , q).

Formulation of the problem in any dimension
Let R = C[x 0 , x 1 , · · · , x n−1 ] be the polynomial algebra with n generators. For given n − 2 polynomials P 1 , P 2 , · · · , P n−2 ∈ R, one can associate the JPS π(P 1 , · · · , P n−2 ) on R given by the formula: for f, g ∈ R.
We will denote by P the particular Casimir P = n−2 i=1 P i of the Poisson structure π(P 1 , · · · , P n−2 ). We suppose that each P i is homogeneous in the sense of τ -degree.
Proof. Let i < j ∈ Z/nZ and consider the set I i,j , formed by the integers i 1 < i 2 < · · · < i n−2 ∈ Z/nZ \ {i, j}. We denote by S i,j the set of all permutation of elements of I { i, j}. We have: .
And we obtain the first part of the result. The second part is the direct consequence of facts that α(i 1 ) + 1 = · · · = α(i n−2 ) + 1 ∈ Z/nZ \ {i + 1, j + 1} and the τ -degree condition.
There are two approaches to determine the dimension of H nr .
Hence, the dimension H nr is just the number of nonnegative integer points contain in the polytope given by the system (21), where r = s + 1.
In dimension 4, we get the following polytope: For the second method, one can observe that the dimension of H nr) is nothing else that the cardinality of the set S C of all compositions (s 0 , · · · , s n−1 ) of N = (n−1)n 2 r − l subjected to the constraints (19). Therefore if S C is the set of all nonnegative integers (s 0 , · · · , s n−1 ) satisfying the constraints (19) and F C is the associated generating function, then the dimension of H nr is the coefficient of q N in F C (q, q, · · · , q). The set S C consists of all nonnegative integers points contained in the polytope of R n :