Bivariate Poincaré Series for the Algebra of Covariants of a Binary Form

Abstract

A formula for computation of the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡) for the algebra of covariants of binary 𝑑-form is found.

1. Introduction

Let 𝑉_𝑑 be the complex vector space of binary forms of degree 𝑑 endowed with the natural action of the special linear group 𝐺=SL(2,ℂ). Consider the corresponding action of the group 𝐺 on the coordinate rings ℂ[𝑉_𝑑] and ℂ[𝑉_𝑑⊕ℂ²]. Denote by ℐ_𝑑=ℂ[𝑉_𝑑]^𝐺 and by 𝒞_𝑑=ℂ[𝑉_𝑑⊕ℂ²]^𝐺 the subalgebras of 𝐺-invariant polynomial functions. In the language of classical invariant theory, the algebras ℐ_𝑑 and 𝒞_𝑑 are called the algebra of invariants and the algebra of covariants for the binary form of degree 𝑑, respectively. The algebra 𝒞_𝑑 is a finitely generated bigraded algebra: 𝒞_𝑑=𝒞_𝑑_0,0+𝒞_𝑑_1,0+⋯+𝒞_𝑑_𝑖,𝑗+⋯,(1.1) where each subspace (𝒞_𝑑)_𝑖,𝑗 of covariants of degree 𝑖 and order 𝑗 is finite dimensional. The formal power series 𝒫_𝑑(𝑧,𝑡)∈ℤ[[𝑧,𝑡]],𝒫_𝑑(𝑧,𝑡)=^∞_𝑖_,𝑗=0dim𝒞_𝑑_𝑖,𝑗𝑧^𝑖𝑡^𝑗,(1.2) is called the bivariate Poincaré series of the algebra of covariants 𝒞_𝑑. It is clear that the series 𝒫_𝑑(𝑧,0) is the Poincaré series of the algebra ℐ_𝑑 and the series 𝒫_𝑑(𝑧,1) is the Poincaré series of the algebra 𝒞_𝑑 with respect to the usual grading of the algebras under degree. The finitely generation of the algebra of covariants implies that its bivariate Poincaré series is the power series expansion of a rational function of two variables 𝑧,𝑡. We consider here the problem of computing efficiently this rational function.

Calculating the Poincaré series of the algebras of invariants and covariants was an important object of research in invariant theory in the 19th century. For the cases 𝑑≤10, 𝑑=12 the series 𝒫_𝑑(𝑧,𝑡) were calculated by Sylvester, see in [1, 2] the big tables of 𝒫_𝑑(𝑧,𝑡), named them as generating functions for covariants, reduced form. All those calculations are correct up to 𝑑=6.

Relatively recently, Springer [3] found an explicit formula for computing the Poincaré series of the algebra of invariants ℐ_𝑑. In the paper we have proved a Cayley-Sylvester-type formula for calculating of dim(𝒞_𝑑)_𝑖,𝑗 and a Springer-type formula for calculation of 𝒫_𝑑(𝑧,𝑡). By using the formula, the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡) is calculated for 𝑑≤20.

2. Cayley-Sylvester-Type Formula for dim(𝒞_𝑑)_𝑖,𝑗

To begin, we give a proof of the Cayley-Sylvester-type formula for the dimension of the graded component (𝒞_𝑑)_𝑖,𝑗.

Let 𝑉_𝑑=⟨𝑣₀,𝑣₁,…,𝑣_𝑑⟩ and dim𝑉_𝑑=𝑑+1 be standard irreducible representation of the Lie algebra 𝔰𝔩₂. The basis elements ⁰¹⁰⁰,⁰⁰¹⁰, ¹⁰⁰⁻¹ of the algebra 𝔰𝔩₂ act on 𝑉_𝑑 by the derivations 𝐷₁,𝐷₂,𝐸: 𝐷₁𝑣_𝑖=𝑖𝑣_𝑖−1,𝐷₂𝑣_𝑖=(𝑑−𝑖)𝑣_𝑖+1,𝐸𝑣_𝑖=(𝑑−2𝑖)𝑣_𝑖.(2.1) The action of 𝔰𝔩₂ is extended to an action on the symmetrical algebra 𝑆(𝑉_𝑑) in the natural way.

Let 𝔲₂ be the maximal unipotent subalgebra of 𝔰𝔩₂. The algebra 𝒮_𝑑, defined by 𝒮_𝑑∶=𝑆𝑉_𝑑^𝔲2=𝑣∈𝑆𝑉_𝑑∣𝐷₁(𝑣)=0,(2.2) is called the algebra of semi-invariants of the binary form of degree 𝑑. For any element 𝑣∈𝒮_𝑑, a natural number 𝑠 is called the order of the element 𝑣 if the number 𝑠 is the smallest natural number such that 𝐷^𝑠2(𝑣)≠0,𝐷^2𝑠+1(𝑣)=0.(2.3) It is clear that any semi-invariant 𝑣∈𝒮_𝑑 of order 𝑖 is the highest weight vector for an irreducible 𝔰𝔩₂-module of the dimension 𝑖+1 in 𝑆(𝑉_𝑑).

The classical theorem [4] of Roberts implies an isomorphism of the algebra of covariants and the algebra of semi-invariants. Furthermore, the order is preserved through the isomorphism. Thus, it is enough to compute the Poincaré series of the algebra 𝒮_𝑑.

The algebra 𝑆(𝑉_𝑑) is ℕ-graded 𝑆𝑉_𝑑=𝑆⁰𝑉_𝑑+𝑆¹𝑉_𝑑+⋯+𝑆^𝑖𝑉_𝑑+⋯,(2.4) and each 𝑆^𝑖(𝑉_𝑑) is a completely reducible representation of the Lie algebra 𝔰𝔩₂. Thus, the following decomposition holds 𝑆^𝑖𝑉_𝑑≅𝛾_𝑑(𝑖,0)𝑉₀+𝛾_𝑑(𝑖,1)𝑉₁+⋯+𝛾_𝑑(𝑖,𝑑⋅𝑛)𝑉_𝑑⋅𝑖,(∗) here 𝛾_𝑑(𝑖,𝑗) is the multiplicity of the representation 𝑉_𝑗 in the decomposition of 𝑆^𝑖(𝑉_𝑑). On the other hand, the multiplicity 𝛾_𝑑(𝑖,𝑗) of the representation 𝑉_𝑗 is equal to the number of linearly independent homogeneous semi-invariants of degree 𝑖 and order 𝑗 for the binary 𝑑-form. This argument proves the following.

Lemma 2.1. dim𝒞_𝑑_𝑖,𝑗=𝛾_𝑑(𝑖,𝑗).(2.5)

The set of weights (eigenvalues of the operator 𝐸) of a representation 𝑊 denote by Λ_𝑊, in particular, Λ_𝑉𝑑={−𝑑,−𝑑+2,…,𝑑−2,𝑑}.

A formal sum Char(𝑊)=_{𝑘∈Λ𝑊}𝑛_𝑊(𝑘)𝑞^𝑘,(2.6) is called the character of a representation 𝑊, here 𝑛_𝑊(𝑘) denotes the multiplicity of the weight 𝑘∈Λ_𝑊. Since, the multiplicity of any weight of the irreducible representation 𝑉_𝑑 is equal to 1, we have Char𝑉_𝑑=𝑞^−𝑑+𝑞^−𝑑+2+⋯+𝑞^𝑑−2+𝑞^𝑑.(2.7)

The character Char(𝑆^𝑛(𝑉_𝑑)) of the representation 𝑆^𝑛(𝑉_𝑑) equals 𝐻_𝑛𝑞^−𝑑,𝑞^−𝑑+2,…,𝑞^𝑑,(2.8) (see [5]), where 𝐻_𝑛(𝑥₀,𝑥₁,…,𝑥_𝑑) is the complete symmetrical function 𝐻_𝑛𝑥₀,𝑥₁,…,𝑥_𝑑=_|𝛼|=𝑛𝑥^𝛼0₀𝑥^𝛼1₁⋯𝑥^𝛼𝑑_𝑑,|𝛼|=_𝑖𝛼_𝑖.(2.9)

By replacing 𝑥_𝑘 with 𝑞^𝑑−2𝑘, 𝑘=0,…,𝑑, we obtain the specialized expression for Char(𝑆^𝑛(𝑉_𝑑)): Char𝑆^𝑛𝑉_𝑑=_|𝛼|=𝑛𝑞^𝑑^𝛼0𝑞^𝑑−2⋅1^𝛼1⋯𝑞^𝑑−2𝑑^𝛼𝑑=_|𝛼|=𝑛𝑞^{𝑑𝑛−2(𝛼1+2𝛼2+⋯+𝑑𝛼𝑑)}=^𝑑𝑛_𝑘=_−𝑑𝑛𝜔_𝑑𝑛,𝑑𝑛−𝑘2𝑞^𝑘,(2.10) here 𝜔_𝑑(𝑛,(𝑑𝑛−𝑘)/2) is the number nonnegative integer solutions of the equation 𝛼₁+2𝛼₂+⋯+𝑑𝛼_𝑑=𝑑𝑛−𝑘2,(2.11) on the assumption that 𝛼₀+𝛼₁+⋯+𝛼_𝑑=𝑛. In particular, the coefficient of 𝑞⁰ (the multiplicity of zero weight ) is equal to 𝜔_𝑑(𝑛,𝑑𝑛/2), and the coefficient of 𝑞¹ is equal to 𝜔_𝑑(𝑛,(𝑑𝑛−1)/2).

On the other hand, the decomposition (*) implies the equality for the characters: Char𝑆^𝑛𝑉_𝑑=𝛾_𝑑(𝑛,0)Char𝑉₀+𝛾_𝑑(𝑛,1)Char𝑉₁+⋯+𝛾_𝑑(𝑛,𝑑𝑛)Char𝑉_𝑑𝑛.(2.12) We can summarize what we have shown so far in the following.

Theorem 2.2. dim𝒞_𝑑_𝑖,𝑗=𝜔_𝑑𝑖,𝑑𝑖2−𝑗−𝜔^𝑑𝑖,𝑑𝑖−(2𝑗+2).(2.13)

Proof. The weight 𝑗 appears once in any representation 𝑉_𝑘, for 𝑘=𝑗mod2,𝑘≥𝑗. Therefore 𝜔_𝑑𝑖,𝑑𝑖2−𝑗=𝛾^𝑑(𝑖,𝑗)+𝛾_𝑑(𝑖,𝑗+2)+⋯+𝛾_𝑑(𝑖,𝑗+4)+⋯.(2.14) Similarly, 𝜔_𝑑𝑖,𝑑𝑖−(2𝑗+2)=𝛾^𝑑(𝑖,𝑗+2)+𝛾_𝑑(𝑖,𝑗+4)+⋯+𝛾_𝑑(𝑖,𝑗+6)+⋯.(2.15) Thus, 𝜔_𝑑𝑖,𝑑𝑖2−𝑗−𝜔^𝑑𝑖,𝑑𝑖−(2𝑗+2)=𝛾^𝑑(𝑖,𝑗).(2.16) By using Lemma 2.1 we obtain dim𝒞_𝑑_𝑖,𝑗=𝜔_𝑑𝑖,𝑑𝑖2−𝑗−𝜔^𝑑𝑖,𝑑𝑖−(2𝑗+2).(2.17)

For another proof of the formula see [3].

Note that the original Cayley-Sylvester formula is dimℐ_𝑑_𝑛=𝜔_𝑑𝑛,𝑑𝑛2−𝜔^𝑑𝑛,𝑑2𝑛−1.(2.18) Also, in [6] we proved that dim𝒞_𝑑_𝑛=𝜔_𝑑𝑛,𝑑𝑛2+𝜔^𝑑𝑛,𝑑𝑛2−1.(2.19) Here (ℐ_𝑑)_𝑛,(𝒞_𝑑)_𝑛 are the components of standard grading of the algebras ℐ_𝑑, 𝒞_𝑑 under degree.

3. Calculation of dim(𝐶_𝑑)_𝑖,𝑗

It is well known that the number 𝜔_𝑑(𝑖,(𝑑𝑖−𝑗)/2) of nonnegative integer solutions of the following system 𝛼₁+2𝛼₂+⋯+𝑑𝛼_𝑑=𝑑𝑖−𝑗𝛼₀2,+𝛼₁+⋯+𝛼_𝑑=𝑖,(3.1) is given by the coefficient of 𝑧^𝑛𝑡^{(𝑑𝑖−𝑗)/2} of the generating function𝑓_𝑑(𝑧,𝑡)=1(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡^𝑑.(3.2) We will use the notation [𝑥^𝑘]𝐹(𝑥) to denote the coefficient of 𝑥^𝑘 in the series expansion of 𝐹(𝑥)∈ℂ[[𝑥]]. Thus 𝜔_𝑑𝑖,𝑑𝑖2−𝑗=𝑧^𝑖𝑡^{(𝑑𝑖−𝑗)/2}𝑓_𝑑(𝑧,𝑡).(3.3) It is clear that 𝜔_𝑑𝑖,𝑑𝑖2−𝑗=𝑧^𝑖𝑡^{𝑑𝑖−𝑗}𝑓_𝑑𝑧,𝑡²=𝑧𝑡^𝑑^𝑖𝑡^𝑗𝑓_𝑑𝑧,𝑡².(3.4)

Similarly, the number 𝜔_𝑑(𝑖,(𝑑𝑖−(𝑗+2))/2) of nonnegative integer solutions of the following system𝛼₁+2𝛼₂+⋯+𝑑𝛼_𝑑=𝑑𝑖−(𝑗+2)𝛼₀2,+𝛼₁+⋯+𝛼_𝑑=𝑖,(3.5)

equals 𝑧^𝑖𝑡^{(𝑑𝑖−(𝑗+2))/2}𝑓_𝑑(𝑧,𝑡)=𝑧^𝑖𝑡^{𝑑𝑖−(𝑗+2)}𝑓_𝑑𝑧,𝑡²=𝑧𝑡^𝑑^𝑖𝑡^𝑗+2𝑓_𝑑𝑧,𝑡².(3.6)

Therefore, 𝜔_𝑑=𝑖,𝑑𝑖𝑧2−𝑡_𝑑𝑗−𝜔^𝑑𝑖,𝑑𝑖−(2𝑗+2)=𝑧𝑡_𝑑^𝑖𝑡^𝑗𝑓_𝑑𝑧,𝑡²−𝑧𝑡^𝑑^𝑖𝑡^𝑗+2𝑓_𝑑𝑧,𝑡²=𝑧_𝑖^𝑖𝑡^𝑗−𝑡^𝑗+2𝑓_𝑑𝑧,𝑡²=𝑧𝑡^𝑑^𝑖𝑡^𝑗1−𝑡²𝑓_𝑑𝑧,𝑡²𝑡^{𝑑𝑖−𝑗}1−𝑡²𝑓_𝑑𝑧,𝑡².(3.7) Thus, the following statement holds.

Theorem 3.1. The number dim(𝒞_𝑑)_𝑖,𝑗 of linearly independent covariants of degree 𝑖 and order 𝑗 for the binary d- form is given by the formula dim𝐶_𝑑_𝑖,𝑗=𝑧^𝑖𝑡^{𝑑𝑖−𝑗}1−𝑡²(1−𝑧)1−𝑧𝑡²⋯1−𝑧𝑡^2𝑑.(3.8)
It is clear that 𝑧^𝑖𝑡^{𝑑𝑖−𝑗}1−𝑡²=𝑧_𝑖(1−𝑧)1−𝑧𝑡²⋯1−𝑧𝑡^2𝑑𝑡^{(𝑑𝑖−𝑗)/2}1−𝑡(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡^𝑑.(3.9) By using the decomposition 1(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡^𝑑=^∞_𝑘=0𝑑𝑖_𝑡𝑧^𝑖,(3.10) where ^𝑑𝑛_𝑞 is the 𝑞-binomial coefficient 𝑑𝑛_𝑞∶=1−𝑞^𝑑+11−𝑞^𝑑+2⋯1−𝑞^𝑑+𝑛(1−𝑞)1−𝑞²⋯(1−𝑞^𝑛),(3.11) one obtains the well-known formula dim𝐶_𝑑_𝑖,𝑗=𝑡^{(𝑑𝑖−𝑗)/2}(1−𝑡)𝑑𝑖_𝑡,(3.12) for instance, see [3].

4. Explicit Formula for 𝒫_𝑑(𝑧,𝑡)

Let us prove Springer-type formula for the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡) of the algebra covariants of the binary 𝑑-form. Consider the ℂ-algebra ℤ[[𝑡,𝑧]] of formal power series. For an integer 𝑑∈ℕ define the ℂ-linear function Ψ_𝑑∶ℤ[[𝑧,𝑡]]⟶ℤ[[𝑧,𝑡]],(4.1) in the following way Ψ_𝑑𝑧^𝑖𝑡^𝑗=𝑧^𝑖0,𝑡^{𝑑𝑖−𝑗},iiff𝑑𝑑𝑖𝑖−−𝑗𝑗<≥00.,(4.2)

The main idea of the ensuing calculations is that the Poincaré series 𝒫_𝑑(𝑧,𝑡) can be expressed in terms of function Ψ_𝑑. The following simple but important statement holds.

Lemma 4.1. 𝒫_𝑑(𝑧,𝑡)=Ψ_𝑑(1−𝑧𝑡)1−1𝑧𝑡−²𝑡²⋯1−𝑧𝑡^2𝑑.(4.3)

Proof. Theorem 2.2 implies that dim(𝐶_𝑑)_𝑖,𝑗=[𝑧^𝑖𝑡^{𝑑𝑖−𝑗}]𝑓_𝑑(𝑧,𝑡²). Then 𝒫_𝑑(𝑧,𝑡)=^∞_𝑖_,𝑗=0dim𝐶_𝑑_𝑖,𝑗𝑧^𝑖𝑡^𝑗=^∞_𝑖_,𝑗=0𝑧^𝑖𝑡^{𝑑𝑖−𝑗}𝑓_𝑑𝑧,𝑡²𝑧^𝑖𝑡^𝑗=Ψ_𝑑𝑓_𝑑𝑧,𝑡².(4.4)

Let 𝜓_𝑛∶ℤ[[𝑡]]→ℤ[[𝑡,𝑧]],𝑛∈ℤ be a ℂ-linear function defined by 𝜓_𝑛(𝑡^𝑚)∶=𝑧^𝑖𝑡^𝑗,where𝑖∶=min𝑘^∣𝑛𝑘^−𝑚≥0,𝑗=𝑛𝑖−𝑚,(4.5)

for 𝑖,𝑗,𝑚,𝑛∈ℕ. Note that 𝜓_𝑛(𝑡⁰)=1 and 𝜓₁(𝑡^𝑚)=𝑧^𝑚, 𝜓₀(𝑡^𝑚)=1. Also, put 𝜓_𝑛(𝑡^𝑚)=0 for 𝑛<0. It is clear that 𝜓_𝑛(𝑡^{𝑛𝑖−𝑗})=𝑧^𝑖𝑡^𝑗 if 𝑛𝑖−𝑗≥0, 𝑗<𝑛.

In important special cases, calculating the functions Ψ can be reduced to calculating the functions 𝜓. The following statements hold.

Lemma 4.2. (i) For 𝑅(𝑡),𝐻(𝑡)∈ℂ[[𝑡]] holds 𝜓_𝑛(𝑅(𝑡^𝑛)𝐻(𝑡))=𝑅(𝑧)𝜓_𝑛(𝐻(𝑡)).
(ii) For 𝑅(𝑡)∈ℂ[[𝑡]] and for 𝑛,𝑘∈ℕ holds Ψ_𝑛1𝑅−(𝑧𝑡)𝑡^𝑘=⎧⎪⎨⎪⎩𝜓^𝑛−𝑘(𝑅(𝑡))1−𝑧𝑡^𝑛−𝑘0,,if𝑛𝑛≥<𝑘𝑘,.(4.6)

Proof. (i) The statement follows from the linearity of the function 𝜓_𝑛 and from the following simple observation: 𝜓_𝑛𝑡^𝑛𝑘𝑡^{𝑛𝑖−𝑗}=𝜓_𝑛𝑡^{𝑛(𝑘+𝑖)−𝑗}=𝑧^𝑘+𝑖𝑡^𝑗=𝑧^𝑘𝑧^𝑖𝑡^𝑗=𝑧^𝑘𝜓_𝑛𝑡^{𝑛𝑖−𝑗},(4.7) for 𝑛𝑖−𝑗≥0 and 𝑗<𝑛.
(ii) Let 𝑅(𝑡)=∑^∞𝑚=0𝑎_𝑚𝑡^𝑚. Then for 𝑘<𝑛 we have Ψ_𝑛1𝑅−(𝑧𝑡)𝑡^𝑘=Ψ^𝑛^∞_𝑚,𝑠=0𝑎_𝑚𝑡^𝑚𝑧𝑡^𝑘^𝑠=Ψ^𝑛^∞_𝑚,𝑠=0𝑎_𝑚𝑧^𝑠𝑡^{𝑘𝑠+𝑚}=_{(𝑛−𝑘)𝑠−𝑚≥0}𝑎_𝑚𝑧^𝑠𝑡^{(𝑛−𝑘)𝑠−𝑚}=^∞_𝑚_,𝑠=0𝑎_𝑚𝜓_𝑛−𝑘(𝑡^𝑚)𝑧𝑡^𝑛−𝑘^𝑠=^∞_𝑚₌₀𝑎_𝑚𝜓_𝑛−𝑘(𝑡^𝑚)11−𝑧𝑡^𝑛−𝑘=𝜓^𝑛−𝑘(𝑅(𝑡))1−𝑧𝑡^𝑛−𝑘.(4.8)

Now we can present Springer-type formula for calculating of the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡).

Theorem 4.3. 𝒫_𝑑(𝑧,𝑡)=_{0≤𝑘<𝑑/2}𝜓_𝑑−2𝑘(−1)^𝑘𝑡^𝑘(𝑘+1)1−𝑡²𝑡²,𝑡²_𝑘𝑡²,𝑡²_𝑑−𝑘1−𝑧1𝑡^𝑑−2𝑘,(4.9) here (𝑎,𝑞)_𝑛=(1−𝑎)(1−𝑎𝑞)⋯(1−𝑎𝑞^𝑛−1) is 𝑞-shifted factorial.

Proof. Consider the partial fraction decomposition of the rational function 𝑓_𝑑(𝑧,𝑡²): 𝑓_𝑑𝑧,𝑡²=^𝑑_𝑘=0𝑅_𝑘(𝑧)1−𝑡𝑧^2𝑘.(4.10) It is easy to see, that 𝑅_𝑘(𝑡)=lim_{𝑧→𝑡−2𝑘}𝑓_𝑑𝑧,𝑡²1−𝑧𝑡^2𝑘=lim_{𝑧→𝑡−2𝑘}1−𝑡²=1−𝑡₂(𝑧,𝑡)_𝑑+11−𝑧𝑡^2𝑘=1−𝑡^−2𝑘1−𝑡𝑡^2−2𝑘⋯1−_{2𝑘+(2𝑘−2)+⋯+2}𝑡^{2(𝑘−1)−2𝑘}1−𝑡^{2(𝑘+1)−2𝑘}⋯1−𝑡^{2𝑑−2𝑘}1−𝑡²=(−𝑡1^2𝑘)−_𝑘1𝑡^2𝑘−2−1⋯𝑡²−11−𝑡²⋯1−𝑡^{2𝑑−2𝑘}𝑡^𝑘(𝑘+1)1−𝑡²𝑡²,𝑡²_𝑘𝑡²,𝑡²_𝑑−𝑘.(4.11) Using the above Lemmas we obtain 𝒫_𝑑(𝑧,𝑡)=Ψ_𝑑𝑓_𝑑𝑧,𝑡²=Ψ_𝑑^𝑛_𝑘=0𝑅_𝑘𝑡²=_{0≤𝑘<𝑑/2}1−𝑧𝑡^2𝑘𝜑_𝑑−2𝑘(−1)^𝑘𝑡^𝑘(𝑘+1)1−𝑡²𝑡²,𝑡²_𝑘𝑡²,𝑡²_𝑑−𝑘1−𝑧1𝑡^𝑑−2𝑘.(4.12)

Corollary 4.4. A denominator of the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡), 𝑑>2 can be written in the form ^[](^𝑑+1)/2_𝑘=01−𝑧𝑡^𝑑−2𝑘^𝑑−2_𝑖=11−𝑧^𝑘𝑖,(4.13) where 𝑘₁,𝑘₂,…,𝑘_𝑑−2 are the degrees of elements of homogeneous system of parameters for the algebra of invariants ℐ_𝑑.

Proof. The formula of Theorem 4.3 implies that the bivariate Poincaré series has the form 𝒫_𝑑(𝑧,𝑡)=𝑃^𝑑(𝑧,𝑡)∏^{[(𝑑+1)/2]𝑘=0}1−𝑧𝑡^𝑑−2𝑘𝑅_𝑑(𝑧),(4.14) for some polynomials 𝑃_𝑑(𝑧,𝑡),𝑅_𝑑(𝑧). Thus, the Poincaré series for the algebra of invariants ℐ_𝑑 has the form 𝒫_𝑑(𝑧,0)=𝑃^𝑑(𝑧,0)𝑅_𝑑(𝑧).(4.15) The algebra of invariants ℐ_𝑑 is Cohen-Macaulay, and its transcendence degree for 𝑑>2 equals 𝑑−2, see [3]. Therefore it has a homogeneous system of 𝑑−2 parameters and the denominator 𝑅_𝑑(𝑧) of its Poincaré series can be written in the following way 𝑅_𝑑(𝑧)=(1−𝑧^𝑘1)(1−𝑧^𝑘2)⋯(1−𝑧^𝑘𝑑−2), where 𝑘₁,𝑘₂,…,𝑘_𝑑−2 are the degrees of elements of this homogeneous system of parameters.

5. Examples

For direct computations we use the following technical lemma.

Lemma 5.1. For 𝑅(𝑡)∈ℂ[[𝑡]] one has 𝜓_𝑛1−𝑡^𝑘11−𝑅𝑡(𝑡^𝑘2)⋯1−𝑡^𝑘𝑚=𝜓^𝑛𝑅(𝑧)𝑄_𝑛𝑡^𝑘1𝑄_𝑛𝑡^𝑘2𝑄_𝑛𝑡^𝑘𝑚1−𝑧^𝑘11−𝑧^𝑘2⋯1−𝑧^𝑘𝑚,(5.1) here 𝑄_𝑛(𝑡)=1+𝑡+𝑡²+⋯+𝑡^𝑛−1, and 𝑘_𝑖 are natural numbers.

Proof. Taking into account Lemma 4.2 we get 𝜓_𝑛1𝑔−(𝑡𝑡)^𝑚=𝜓^𝑛1𝑔−(𝑡𝑡)^𝑛𝑚1−𝑡^𝑛𝑚1−𝑡^𝑚=1−1𝑡^𝑚𝜓_𝑛=1𝑔(𝑡)11−−𝑡𝑡^𝑛𝑚𝑚1−𝑡^𝑚=1𝜓_𝑛𝑔(𝑡)1+𝑡^𝑚+(𝑡^𝑚)²+⋯+(𝑡^𝑚)^𝑛−11−𝑡^𝑚𝜓_𝑛𝑔(𝑡)𝑄_𝑛(𝑡^𝑚).(5.2) In a similar fashion we prove the general case.

By using Lemma 5.1 the bivariate Poincaré series 𝒫_𝑑(𝑧,𝑡) for 𝑑≤20 are found. All these results agree with Sylvester's calculations up to 𝑑=6, see [1, 2].

Below is the list of several series:𝒫₁(𝑧,𝑡)=11−𝑧𝑡,𝒫²(𝑧,𝑡)=1𝒫₃1−𝑧𝑡²1−𝑧²,(𝑧,𝑡)∶=𝑧²𝑡²−𝑧𝑡+1𝒫₄(1−𝑧𝑡)1−𝑧𝑡³1−𝑧⁴,(𝑧,𝑡)=𝑧²𝑡⁴−𝑧𝑡²+1𝒫₅1−𝑧𝑡²1−𝑡⁴𝑧1−𝑧²1−𝑧³,(𝑧,𝑡)=𝑝⁵(𝑧,𝑡)𝑝₅(1−𝑧𝑡)1−𝑡³𝑧1−𝑧𝑡⁵1−𝑧⁸1−𝑧⁶1−𝑧⁴,(𝑧,𝑡)=1+𝑧⁷𝑡³−𝑧⁶𝑡⁴+𝑧²𝑡²+2𝑧⁷𝑡−𝑧⁵𝑡⁵−𝑧⁸𝑡²−2𝑧⁸𝑡⁶−𝑧⁸𝑡⁴+𝑧⁵𝑡³+𝑧⁵𝑡+𝑧⁹𝑡⁷−𝑧¹⁰𝑡⁶−+𝑧𝑧¹⁰₁₃𝑡²−𝑧¹⁰𝑡⁴−𝑧¹¹𝑡³+𝑧⁹𝑡³−𝑡³𝑧−𝑧⁶+𝑧⁴𝑡⁴−𝑧𝑡+𝑧²𝑡⁶+𝑧²𝑡⁴+𝑧¹²+𝑧¹⁴𝑡⁶−𝑧¹³𝑡𝑡⁵−𝑧¹³𝑡³−𝑧¹⁵𝑡⁷+𝑧¹⁴𝑡⁴−𝑧³𝑡⁷+𝑧⁷𝑡⁵.(5.3)