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

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Volume 2011, Article ID 252186,18pages doi:10.1155/2011/252186

Research Article

Integer Solutions of Integral Inequalities and H -Invariant Jacobian Poisson Structures

G. Ortenzi,

¹

V. Rubtsov,

²

and S. R. Tagne Pelap

³

1Dipartimento di Matematica Pura e Applicazioni, Universit`a degli Milano Bicocca, Via R.Cozzi 53, 20125 Milano, Italy

2Département de Mathématiques 2, Laboratoire Angevin de Recherche en Mathématiques Université D’Angers, boulevard Lavoisier, 49045 Angers, France

3Mathematics Research Unit at Luxembourg, University of Luxembourg, 6 rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg

Correspondence should be addressed to S. R. Tagne Pelap,serge.pelap@uni.lu Received 14 April 2011; Accepted 3 June 2011

Academic Editor: Yao-Zhong Zhang

Copyrightq2011 G. Ortenzi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

This paper continues the authors’ program of studies of the Heisenberg invariance properties of polynomial Poisson algebras which were started in1and extended in2,3. Formally speaking, we consider the polynomials innvariablesCx0, x1, . . . , x_n−1overCand the action of some subgroupH_nofGL_nCgenerated by the shifts operatorsx_i → x_i1modZnand by the operatorsx → εⁱx_i, whereεⁿ1. We are interested in the polynomial Poisson brackets on Cx0, x1, . . . , x_n−1which are “stable” under this actionswe 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 “quasiclassical 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

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provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines anH_n-action on the elliptic quadratic Poisson structures onPⁿ. The latter are identified with Poisson structures on some moduli spaces of the degree nand rank k1 vector bundles with parabolic structurethe flag 0⊂ F ⊂C^k1on the elliptic curveE. We will denote this elliptic Poisson algebras byqn;kE. The algebrasqn;kEarise in the Feigin- Odesskii “deformational” approach and form a subclass of polynomial Poisson structures.

A comprehensive review of elliptic algebras can be found in 4to which we refer for all additional information. We will mention only that as we have proved in3all elliptic Poisson algebrasbeing in particular Heisenberg-invariantare 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 Qis given by the determinant of Jacobi matrix of functions P, Q, P1, . . . , P_n−2. The polynomials Pi, 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 Cx0, . . . , x_n−1. This type of Poisson algebras was intensively studieddue to their natural origin and relative simplicityin a huge number of publications among which we should mention1,5–9.

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-Taten3and Sklyaninn4algebras which are both elliptic and Jacobian. It is no longer true forn >4. The relations between the Sklyanin Poisson algebrasqn,kEwhose centrum has dimension 1fornoddand 2fornevenin the casek 1 and is generated byl gcdn, k 1Casimirs for q_n,kEfor 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 8 and was attributed to so-called Briesckorn-Pham polynomials for n5

P1⁴

i0

αixi; P2⁴

i0

βix_i²; P3⁴

i0

γix²_i. 1.1

It is easy to check that the homogeneous quinticP P₁P₂P₃see Section4.2defines a Casimir for some rational degeneration ofone ofelliptic algebrasq_5,1Eandq_5,2Eif 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 subvector 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 HiofHof degreeiis in bijection with integer solutions of a system of Diophant inequalities.

Geometric interpretation of the dimension ofHi is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3,His 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 theH-invariant not necessarily quadratic Jacobian Poisson algebras with three generators.

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As a byproduct, we explicitly compute the Poincar´e series ofH. In this dimension, we observe that theH-invariant JPS of degree 5 is given by the Casimir sextic

P^∨ 1 6a

x⁶₀x⁶₁x₂⁶ 1

3b

x₀³x³₁x³₀x³₂x₁³x₂³ c

x⁴₀x₁x₂x₀x₁⁴x₂x₀x₁x⁴₂ 1

2dx²₀x²₁x²₂,

1.2

a, b, c, d∈C. This structure is a “projectively dual” to the Artin-Schelter-Tate elliptic Poisson structure which is theH-invariant JPS given by the cubic

P

x³₀x³₁x³₂

γx₀x₁x₂, 1.3

whereγ ∈C. In fact, the algebraic varietyE^∨ :{P^∨ 0} ∈P²is thegenericallyprojectively dual to the elliptic curveE:{P0} ⊂P².

The paper is organized as follows: in Section 2, 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 Section3, 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.

2. Preliminary Facts

Throughout of this paper,Kis a field of characteristic zero. Let us start by remembering some elementary notions of the Poisson geometry.

2.1. Poisson Algebras and Poisson Manifold

Let Rbe a commutative K-algebra. One says thatRis a Poisson algebra if R is endowed with a Lie bracket, indicated with{·,·}, which is also a biderivation. One can also say that Ris endowed with a Poisson structure, and therefore, the bracket{·,·}is called the Poisson bracket. Elements of the center are called Casimirs:a ∈ Ris a Casimir if{a, b} 0 for all b∈ R.

A Poisson manifoldMsmooth, algebraic, etc.is a manifold whose function algebra AC^∞M, regular, etc.is endowed with a Poisson bracket.

As examples of Poisson structures let us consider a particular subclass of Poisson structures which are uniquely characterized by their Casimirs. In the dimension 4, let

q₁ 1 2

x₀²x²₂

kx₁x₃,

q2 1 2

x²₁x²₃

kx0x2

2.1

be two elements ofCx0, x₁, x₂, x₃, wherek∈C.

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OnCx0, x1, x2, x3, a Poisson structureπis defined by f, g

π : df∧dg∧dq1∧dq2

dx0∧dx1∧dx2∧dx3, 2.2

or more explicitlymodZ4

{xi, x_i1}k²x_ix_i1−x_i2x_i3, {xi, x_i2}k

x²_i3−x²_i1

, i0,1,2,3.

2.3

Sklyanin had introduced this Poisson algebra which carries today his name in a Hamiltonian approach to the continuous and discrete integrable Landau-Lifshitz models 10, 11. 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 due to 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 situationis an elliptic curve E given by two quadrics q1,2. We can think about this curveEas a complete intersection of the coupleq₁ 0,q₂0 embedded inCP³as it was observed in Sklyanin’s initial paper.

A possible generalization one can be obtained consideringn−2 polynomialsQiinKⁿ with coordinatesx_i,i0, . . . , n−1. We can define a bilinear diﬀerential operation

{·,·}:Kx1, . . . , x_n⊗Kx1, . . . , x_n−→Kx1, . . . , x_n, 2.4

by

f, g

df∧dg∧dQ1∧ · · · ∧dQ_n−2

dx₁∧dx₂∧ · · · ∧dx_n , f, g∈Kx1, . . . , xn. 2.5 This operation, which gives a Poisson algebra structure onKx1, . . . , xn, is called a Jacobian Poisson structureJPS, and it is a partial case of more generaln−m-ary Nambu operation given by an antisymmetric n − m-polyvector field introduced by Nambu 6 and was extensively studied by Takhtajan5.

The polynomialsQ_i,i1, . . . , n−2 are Casimir functions for the brackets2.5.

There exists a second generalization of the Sklyanin algebra that we will describe briefly in the next subsectionsee, for details,4.

2.2. Elliptic Poisson AlgebrasqnE, ηandqn,kE, η

We report here this subsection from2for sake of self-consistency.

These algebras, defined by Fe˘ıgin and Odesski˘ı, arise as quasiclassical limits of elliptic associative algebrasQ_nE, ηandQ_n,kE, η 12,13.

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LetΓ ZτZ ⊂ Cbe an integral lattice generated by 1 andτ ∈ C, with Imτ > 0.

Consider the elliptic curveEC/Γand a pointηon this curve.

In their article13, givenk < n, mutually prime, Odesski˘ıand Fe˘ıgin construct an algebra, called elliptic,Qn,kE, η, as an algebra defined byngenerators{xi, i ∈ Z/nZ}and the following relations:

r∈Z/nZ

θ_j−irk−10 θkr

η θ_j−i−r

−ηx_j−rx_ir 0, i /j, i, j∈ Z

nZ, 2.6

whereθαare theta functions13.

These family of algebras has the following properties:

1the center of the algebraQn,kE, η, for genericEandη, is the algebra of polynomial ofmpgcdn, k1variables of degreen/m,

2Q_n,kE,0 Cx1, . . . , x_nis commutative,

3Q_n,n−1E, η Cx1, . . . , xnis commutative for allη, 4Q_n,kE, ηQ_n,kE, η, ifkk≡1modn,

5the mapsxi →x_i1etxi →εⁱxi, whereεⁿ1, define automorphisms of the algebra Q_n,kE, η,

6the algebras Qn,kE, η are deformations of polynomial algebras. The associated Poisson structure is denoted byqn,kE, η,

7among the algebras q_n,kE, η, onlyq₃E, η the Artin-Schelter-Tate algebraand the Sklyanin algebraq4E, ηare Jacobian Poisson structures.

2.3. The Heisenberg Invariant Poisson Structures 2.3.1. The G-Invariant Poisson Structures

LetGbe a group acting on a Poisson algebraR.

Definition 2.1. A Poisson bracket{·,·}onRis said to be aG-invariant ifGacts onRby Poisson automorphisms.

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:

R × R^ϕ^g^×ϕ^g

{·,·}

R × R

{·,·}

R ^ϕ^g R

2.7

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2.3.2. The H-Invariant Poisson Structures

In their paper2, the authors introduced the notion ofH-invariant Poisson structures. That is, a special case of aG-invariant structure whenGin the finite Heisenberg group andRis the polynomial algebra. Let us remember this notion.

LetV be a complex vector space of dimensionnande0, . . . , e_n−1a basis ofV. Take the nth primitive root of unityεe^2πi/n.

Considerσ, τofGLVdefined by

σem e_m−1,

τem ε^me_m. 2.8

The Heisenberg of dimensionnis nothing else that the subspaceH_n ⊂GLVgenerated by σandτ.

From now on, we assume thatV Cⁿ, withx0, x1, . . . , x_n−1as a basis and consider the coordinate ringCx0, x₁, . . . , x_n−1.

Naturally,σandτact by automorphisms on the algebraCx0, x1, . . . , x_n−1as follows:

σ·

αx^α₀⁰x^α₁¹· · ·x^α_n−1ⁿ⁻¹

αx^α₀ⁿ⁻¹x^α₁⁰· · ·x^α_n−1ⁿ⁻², τ·

αx₀^α⁰x^α₁¹· · ·x^α_n−1ⁿ⁻¹

ε^α¹^2α²^{···n−1α}ⁿ⁻¹αx^α₀⁰x^α₁¹· · ·x^α_n−1ⁿ⁻¹. 2.9

We introduced in2the notion ofτ-degree on the polynomial algebraCx0, x₁, . . . , x_n−1. Theτ-degree of a monomialM αx^α₀⁰x^α₁¹· · ·x^α_n−1ⁿ⁻¹is the positive integerα₁2α₂· · · n− 1αn−1∈Z/nZifα /0 and−∞if not. Theτ-degree ofMis denotedτM. Aτ-degree of a polynomial is the highestτ-degree of its monomials.

For simplicity, theH_n-invariance condition will be referred from now on just asH- invariance. AnH-invariant Poisson bracket onACx0, x1, . . . , x_n−1is nothing but a bracket onAwhich satisfy the following:

x_i1, x_j1 σ·

xi, xj

,

τ· x_i, x_j

ε^ij x_i, x_j

, 2.10

for alli, j∈Z/nZ.

Theτinvariance is, in some sense, a “discrete” homogeneity.

Proposition 2.2see2. The Sklyanin-Odesskii-Feigin Poisson algebrasqn,kEareH-invariant Poisson algebras.

Therefore, anH-invariant Poisson structures on the polynomial algebraRincludes as the Sklyanin Poisson algebra or more generally of the Odesskii-Feigin Poisson algebras.

In this paper, we are interested in the intersection of the two classes of generalizations of Artin-Shelter-Tate-Sklyanin Poisson algebras: JPS andH-invariant Poisson structures.

Proposition 2.3see2. If{·,·}is anH-invariant polynomial Poisson bracket, the usual polyno- mial degree of the monomial of{xi, x_j}equals to 2sn,s∈N.

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Proposition 2.4see2. LetP∈ RCx0, . . . , x_n−1. For alli∈ {0, . . . , n−1},

σ· ∂F

∂xi ∂σ·F

∂σ·xi. 2.11

3. H -Invariant JPS in Dimension 3

We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras. LetR Cx0, x1, x2be the polynomial algebra with 3 generators. For everyP ∈ A, we have a JPS πPonRgiven by

x_i, x_j ∂P

∂xk

, 3.1

wherei, j, k∈ Z/3Zis a cyclic permutation of0,1,2. LetHbe the set of allP ∈ Asuch thatπPis anH-invariant Poisson structure.

Proposition 3.1. IfP ∈ His a homogeneous polynomial, thenσ·PPandτP 0.

Proof. Leti, j, k∈Z/3Z³be a cyclic permutation of0,1,2. One has x_i, x_j

∂P

∂xk

, 3.2

x_i1, x_j1 ∂P

∂x_k1. 3.3

Using Proposition2.4, we conclude that for allm∈Z/3Z,∂σP/∂x_m∂P/∂x_m. It gives thatσ·P P.

On the other hand, from3.2, one hasτ−P≡ijkmod 3. And we get the second half of the proposition.

Proposition 3.2. His a subalgebra ofR.

Proof. LetP, Q∈ H. It is clear that for allα, β∈C,αPβQbelongs toH.

Let us denote by{·,·}_Fthe JPS associated with the polynomialF ∈ R. It is easy to verify that{xi, x_j}_{P Q}P{xi, x_j}_QQ{xi, x_j}_P. Therefore, it is clear that theH-invariance condition is verified for the JPS associated to the polynomialP Q.

We endowHwith the usual grading of the polynomial algebraR. ForF, an element of R, we denote byFits usual weight degree. We denote byHithe homogeneous subspace ofHof degreei.

Proposition 3.3. If 3 does not dividei∈N(in other wordsi /3k), thenHi 0.

Proof. First of allH0 C. We suppose now thati /0. LetP ∈ Hi, P /0. Then,P i. It follows from Proposition2.3and the definition of the Poisson brackets that there existss∈N such thatxi, x_j 23s. The result follows from the fact thatxi, x_j P−1.

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SetP cαx^α₀⁰x^α₁¹x₂^α², whereα α0, α1, α2. We suppose thatP 31s. We want to find allα₀, α₁, α₂such thatP ∈ Hand, therefore, the dimensionH_31sasC-vector space.

Proposition 3.4. There exists,s, andssuch that

α₀α₁α₂ 31s, 0α₀α₁2α₂3s, α02α10α2 3s, 2α₀0α₁1α₂3s.

3.4

Proof. This is a direct consequence of Proposition3.3.

Proposition 3.5. The system equation3.4has as solutions the following set:

α04r−2s−s, α₁−2rs2s,

α2rs−s,

3.5

wherer1s,sandslive in the polygon given by the following inequalities inR²: xy3r,

2xy4r,

−x−2y−2r,

−xyr.

3.6

Remark 3.6. Forr 1, one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the CasimirP αx³₀x₁³x³₂ βx₀x₁x₂,α, β ∈C. Suppose thatα /0, then it can take the form

P

x³₀x³₁x³₂

γx0x1x2, 3.7

whereγ∈C. The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following nonalgebraic changes of variables. Let

y₀x₀; y₁x₁x^−1/2₂ ; y₂x^3/2₂ . 3.8

The polynomialP in the coordinatesy0;y1;y2has the form P

y³₀y³₁y₂y₂²

γy₀y₁y₂. 3.9

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The Poisson bracket is also polynomialwhich is not evident at all!and has the same form x_i, x_j

∂P

∂x_k, 3.10

wherei, j, kis the cyclic permutation of0,1,2. This JPS structure is no longer satisfying the Heisenberg invariance condition. But it is invariant with respect the following toric action:

C^∗³×P² → Pgiven by

λ·x0:x1:x2

λ²x0:λx1:λ³x2

. 3.11

Put degy₀ 2; degy₂ 1; degy₂ 3. Then, the polynomialPis also homogeneous iny0;y1;y2and defines an elliptic curveP0 in the weighted projective spaceWP2;1;3.

The similar change of variables

z₀x^−3/4₀ x^3/2₁ ; z₁x^1/4₀ x^−1/2₁ x₂; z₂x₀^3/2 3.12

defines the JPS structure invariant with respect to the torus actionC^∗³×P² → Pgiven by λ·x0:x1:x2

λx0:λx1:λ²x2

, 3.13

and related to the elliptic curve 1/3z²₂z²₀z2z0z³₁ kz0z1z20 in the weighted projective spaceWP1;1;2.

These structures had appeared in 1, their Poisson cohomology was studied by Pichereau14, and their relation to the noncommutative del Pezzo surfaces and Calabi-Yau algebras were discussed in15.

Proposition 3.7. The subset ofR²given by the system3.6is a triangleTr with0, r,r,2r, and 2r,0as vertices. Then, dimH3r CardTr∩N².

Remark 3.8. Forr 2, the case of Figure1, the generic Heisenberg-invariant JPS is given by the sextic polynomial

P^∨ 1 6a

p⁶₀p⁶₁p⁶₂ 1

3b

p³₀p₁³p₀³p³₂p³₁p³₂ c

p⁴₀p₁p₂p₀p⁴₁p₂p₀p₁p⁴₂ 1

2dp²₀p²₁p₂², a, b, c, d∈C.

3.14

The corresponding Poisson bracket takes the form xi, xj

ax⁵_kb

x³_i x³_j

x_k²cxixj

x³_i x³_j4x³_k

dx²_ix²_jxk, 3.15

wherei, j, kare the cyclic permutations of 0,1,2.

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−2 2 4 6 6

4

2

−2 y

x

Figure 1: An example of the triangleT2in the caser2.

This new JPS should be considered as the “projectively dual” to the Artin-Schelter-Tate JPS, since the algebraic varietyE^∨ :P^∨0 is generically the projective dual curve inP²to the elliptic curve

E:P

x³₀x₁³x³₂

γx0x1x20. 3.16

To establish the exact duality and the explicit values of the coeﬃcients, we should use see16, chapter 1Schl¨afli’s formula for the dual of a smooth plane cubicE 0 ⊂P². The coordinatesp0 :p₁ :p₂∈P²^∗of a pointp ∈P²^∗satisfies to the sextic relationE^∨ 0 if and only if the linex0p0x1p1x2p20 is tangent to the conic locusCx, p 0, where

C x, p

0 p0 p1 p2

p0 ∂²P

∂x²₀

∂²P

∂x₀∂x₁

∂²P

∂x₀∂x₂ p1

∂²P

∂x1∂x0

∂²P

∂x1∂x1

∂²P

∂x1∂x2

p₂ ∂²P

∂x2∂x0

∂²P

∂x2∂x1

∂²P

∂x2∂x2

. 3.17

SetSr Tr ∩N².Sr S¹_r ∪ S²_r,S_r¹ {x, y∈ Sr : 0 ≤x≤r}andS¹_r {x, y∈ Sr : r < x≤2r}. dimH3r CardS¹_r CardS_r².

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Proposition 3.9.

Card S_r¹

⎧⎪

⎪⎨

⎪⎪

⎩

3r²6r4

4 if r is even, 3r²6r3

4 if r is odd.

3.18

Proof. Letα∈ {0, . . . , r}, and setD^α_r {x, y∈ S_r¹:xα}. Therefore,

Card S¹_r

^r

α0

CardD^α_r. 3.19

Let

β^α_maxmax β:

α, β

∈ D^α_r , β_min^α min

β: α, β

∈ D_r^α

. 3.20

CardD_r^α β^α_max−β^α_min1.

It is easy to prove that

CardD^α_r α1 α 2

⎧⎪

⎨

⎪⎩ 3α2

2 ifαis even, 3α1

2 ifαis odd.

3.21

The result follows from the summation of all CardD^α_r, α∈ {0, . . . , r}.

Proposition 3.10.

Card S²_r

⎧⎪

⎪⎨

⎪⎪

⎩ 3r²

4 ifr is even, 3r²1

4 ifr is odd.

3.22

Proof. Letα∈ {r1, . . . ,2r}, and setD^α_r {x, y∈ S_r²:xα}. Therefore,

Card S_r²

^2r

αr1

CardD_r^α. 3.23

Let

β^α_maxmax β:

α, β

∈ D^α_r , β^α_minmin

β: α, β

∈ D^α_r

. 3.24

CardD_r^α β^α_max−β^α_min1.

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It is easy to prove that

CardD^α_r 3r1−2αα 2

⎧⎪

⎨

⎪⎩

6r2−3α

2 ifαis even, 6r1−3α

2 ifαis odd.

3.25

The result follows from the summation of all CardD^α_r,α∈ {r1, . . . ,2r}.

Theorem 3.11.

dimH_31s 3 2s²9

2s4. 3.26

Proof. This result is a direct consequence of Propositions3.9and3.10.

Corollary 3.12. The Poincar´e series of the algebrasHis

PH, t

s≥−1

dim

H_31s

t^3s1 1t³t⁶

1−t³3 . 3.27

Remark 3.13. For r 2, the case of Figure 1, our formula gives the same answer like the classical Pick’s formula for integer points in a convex polygonΠwith integer vertices on the plane17, chapter 10

Card

Π∩Z²

AreaΠ 1 2Card

∂Π∩Z²

1. 3.28

Here,s1 and dimH31110. In other hand the Pick’s formula ingredients are

AreaΠ 1 2

2 −4 4 −2

6, 1 2Card

∂Π∩Z²

3, 3.29

hence 63110.

This remark gives a good hint how one can use the developed machinery of integer points computations in rational polytopes to our problems.

4. H -Invariant JPS in Any Dimension

In order to formulate the problem in any dimension, let us remember some number theoretic notions concerning the enumeration of nonnegative integer points in a polytope or more generally discrete volume of a polytope.

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4.1. Enumeration of Integer Solutions to Linear Inequalities

In their papers18,19, 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 sequenceλ λ0, λ2, . . . , λ_n−1of integers to be|λ|λ0· · ·λ_n−1. If sequenceλof weight N has all parts nonnegative, it is called a composition ofN; if, in addition,λis a nonincreasing sequence, we call it a partition ofN.

Given anr×ninteger matrixC ci,j,i, j∈{−1} ∪Z/rZ×Z/nZ, consider the set S_Cof nonnegative integer sequencesλ λ1, λ₂, . . . , λ_nsatisfying the constraints

c_i,−1c_i,0λ₀c_i,1λ₁· · ·c_i,n−1λ_n−10, 0≤r≤n−1. 4.1

The associated full generating function is defined as follows:

F_Cx0, x₂, . . . , x_n−1

λ∈SC

x₀^λ⁰x₁^λ¹· · ·x^λ_n−1ⁿ⁻¹. 4.2

This function “encapsulates” the solution set SC: the coeﬃcient of q^N in FCqx0, qx1, . . . , qx_n−1is a “listing” as the terms of a polynomialof all nonnegative integer solutions to 4.1of weightN, and the number of such solutions is the coeﬃcient ofq^NinF_Cq, q, . . . , q.

4.2. Formulation of the Problem in Any Dimension

Let R Cx0, x₁, . . . , x_n−1 be the polynomial algebra withn generators. For given n−2 polynomialsP₁, P₂, . . . , P_n−2∈ R, one can associate the JPSπP1, . . . , P_n−2onRgiven by

f, g

df∧dg∧dP1∧ · · · ∧dP_n−2

dx₀∧dx₁∧ · · · ∧dx_n−1 , 4.3

forf, g∈ R.

We will denote by P the particular Casimir P _n−2

i1P_i of the Poisson structure πP1, . . . , P_n−2. We suppose that eachP_iis homogeneous in the sense ofτ-degree.

Proposition 4.1. Consider a JPSπP1, . . . , P_n−2given by homogeneous (in the sense ofτ-degree) polynomialsP₁, . . . , P_n−2. IfπP1, . . . , P_n−2isH-invariant, then

τ−σ·P τ−P

⎧⎨

⎩ n

2 ifnis even,

0 ifnis odd, 4.4

whereP P₁P₂· · ·P_n−2.

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Proof. Leti < j ∈Z/nZ, and consider the setIi,j, formed by the integersi1 < i2 <· · ·< i_n−2 ∈ Z/nZ\ {i, j}. We denote byS_i,jthe set of all permutation of elements ofI_i,j. We have

xi, xj

−1^ij−1dx_i∧dx_j∧dP₁∧dP₂∧ · · · ∧dP_n−2 dx₀∧dx₁∧dP₂∧ · · · ∧dx_n−1 −1^ij−1

α∈Si,j

−1^|α| ∂P1

∂x_αi₁· · · ∂P_n−2

∂x_αi_n−2.

4.5

From theτ-degree condition,

ij ≡τ−P1−αi1 · · · τ−Pn−2−αin−2modulon. 4.6

We can deduce, therefore, that

τ−P1· · ·P_n−2≡ nn−1

2 modulon. 4.7

And we obtain the first part of the result. The second part is the direct consequence of facts that

σ· xi, xj

x_i1, x_j1

−1^ij−1

α∈Si,j

−1^|α|∂σ·P1

∂x_αi₁₁ · · ·∂σ·P_n−2

∂x_αi_n−2₁ , 4.8

αi1 1/· · ·/αin−2 1∈Z/nZ\ {i1, j1}and theτ-degree condition.

Set

l

⎧⎨

⎩ n

2 ifnis even, 0 ifnis odd.

4.9

LetHbe the set of allQ∈ Rsuch thatτ−σ·Q τ−Q l. One can easily check the following result.

Proposition 4.2. His a subvector space ofR. It is subalgebra ofRifl0.

We endowHwith the usual grading of the polynomial algebraR. ForQ, an element of R, we denote byQits usual weight degree. We denote byHithe homogeneous subspace ofHof degreei.

Proposition 4.3. Ifnis not a divisor ofi(in other words,i /nm) thenHi0.

Proof. It is clear theH0C. We suppose now thati /0. LetQ∈ Hi,Q /0. Then,

Q

k1,...,ki−1

a_k₁_,...,k_i−1x_k₁· · ·x_k_i−1x_l−k₁_{−···−k}_i−1. 4.10

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Hence,

σ·Q

k1,...,ki−1

ak1,...,k_i−1xk11· · ·xk_i−11x_l−k₁_{−···−k}_i−1₁. 4.11

Sinceτ−σ·Q τ−Q l,i≡0 modulo n.

SetQ βx^α₀⁰x^α₁¹· · ·x^α_n−1ⁿ⁻¹. We suppose that Q n1s. We want to find all α₀, α₁, . . . , α_n−1such thatQ∈ Hand, therefore, the dimensionH_31sasC-vector space.

Proposition 4.4. There exists₀, s₁, . . . , s_n−1such that

α₀α₁· · ·α_n−1 n1s, 0α₀α₁2α₂· · ·n−1α_n−1lns₀, 1α₀2α₁3α₂· · ·n−1α_n−20α_n−1lns₁,

...

n−2α0 n−1α10α₂· · ·n−3α_n−4 n−3α_n−1lns_n−2, n−1α00α₁1α₂· · ·n−3α_n−3 n−2α_n−1lns_n−1.

4.12

Proof. That is, the direct consequence of the fact thatτ−σ·Q τ−Q l.

One can easily obtain the following result.

Proposition 4.5. The system equation4.12has as a solution

αis_n−i−1−s_n−ir, i∈ Z

nZ, 4.13

wherers1 and thes0, . . . , s_n−1satisfy the condition

s0s1· · ·s_n−1 n−1n

2 r−l. 4.14

Thereforeα0, α1, . . . , α_n−1are completely determined by the set of nonnegative integer sequencess0, s₁, . . . , s_n−1satisfying the constraints

c_i:s_n−i−1−s_n−ir≥0, i∈ Z

nZ, 4.15

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and such that

s₀s₁· · ·s_n−1 n−1n

2 r−l. 4.16

There are two approaches to determine the dimension ofHnr.

The first one is exactly as in the case of dimension 3. The constraint4.16is equivalent to say that

s_n−1−s0s₁· · ·s_n−2 n−1n

2 r−l. 4.17

Therefore, by replacings_n−1by this value,α0, . . . , α_n−1 are completely determined by the set of nonnegative integer sequencess0, s₁, . . . , s_n−2satisfying the constraints

c₋₁ :s0s1· · ·s_n−3s_n−2≤ n−1n 2 r−l, c₀: 2s₀s₁· · ·s_n−3s_n−2≤

n−1n

2 1

r−l,

c₁:s₀s₁· · ·s_n−32s_n−2≥

n−1n 2 k1

r−l,

c_i:s_n−i−1−s_n−ir≥0, i∈Z/nZ\ {0,1}.

4.18

Hence, the dimensionHnris just the number of nonnegative integer points contained in the polytope given by the system4.18, wherers1.

In dimension 3, one obtains the triangle inR²given by the verticesA0,2r,Br,2r, andC2r,0 see Section3.

In dimension 4, we get the following polytopesee Figure2.

For the second method, one can observe that the dimension of Hnr is nothing else that the cardinality of the setS_C of all compositions s0, . . . , s_n−1 ofN n−1n/2r − l subjected to the constraints 4.15. Therefore, if SC is the set of all nonnegative integers s0, . . . , s_n−1satisfying the constraints 4.15and F_C is the associated generating function, then the dimension ofHnris the coeﬃcient ofq^NinF_Cq, q, . . . , q. The setS_Cconsists of all nonnegative integers points contained in the polytope ofRⁿ

Pn:

x_n−i−1−x_n−ir≥0, i∈ Z nZ

x_i≥0, i∈ Z

nZ.

4.19

See Figure3.

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10 8 6 4 2 0 0

2 4

6 8

10 10

5

0

Figure 2: An example of the polytopeT4in the caser 4. The vertices are in−1/2, r−1/2,2r −1/2, r−2/3,2r−2/3,3r−2/3,r,2r−1,3r−1,r,2r,3r−2,2r−1/2,3r−1/2,−1/2, and3r−1/2,−1/2, r−1/2.

2 4 6 8 10 12

5 10

0 2 4 6 8 10 12 14

Figure 3: An example of the polytopeP3in the caser2.

Acknowledgments

The authors are grateful to M. Beck and T. Schedler for useful and illuminative discussions.

This work has begun when G. Ortenzi was visiting Mathematics Research Unit at Luxembourg. G. Ortenzi thanks this institute for the invitation and for the kind hospitality.

A Part of this work has been done when S. R. T. Pelap: was visiting Max Planck Institute at Bonn. S. R. T. Pelap thanks this institute for the invitation and for good working conditions.

S. R. T. Pelap have been partially financed by “Fond National de RechercheLuxembourg”.

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He is thankful to LAREMA for a kind invitation and a support during his stay in Angers. V.

Rubtsov was partially supported by the French National Research AgencyANRGrant no.

2011 DIADEMS and by franco-ukrainian PICSCNRS-NASin Mathematical Physics. He is grateful to MATPYL project for a support of T. Schedler visit in Angers and to the University of Luxembourg for a support of his visit to Luxembourg.

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

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