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

1. Introduction

Let 𝑉𝑑 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[𝑉𝑑] and ℂ[𝑉𝑑⊕ℂ2]. Denote by ℐ𝑑=ℂ[𝑉𝑑]𝐺 and by ğ’žğ‘‘=ℂ[𝑉𝑑⊕ℂ2]𝐺 the subalgebras of 𝐺-invariant polynomial functions. In the language of classical invariant theory, the algebras ℐ𝑑 and ğ’žğ‘‘ are called the algebra of invariants and the algebra of covariants for the binary form of degree 𝑑, respectively. The algebra ğ’žğ‘‘ is a finitely generated bigraded algebra: ğ’žğ‘‘=î€·ğ’žğ‘‘î€¸0,0+î€·ğ’žğ‘‘î€¸1,0î€·ğ’ž+⋯+𝑑𝑖,𝑗+⋯,(1.1) where each subspace (ğ’žğ‘‘)𝑖,𝑗 of covariants of degree 𝑖 and order 𝑗 is finite dimensional. The formal power series 𝒫𝑑(𝑧,𝑡)∈ℤ[[𝑧,𝑡]],𝒫𝑑(𝑧,𝑡)=âˆžî“ğ‘–,𝑗=0î‚€î€·ğ’ždim𝑑𝑖,𝑗𝑧𝑖𝑡𝑗,(1.2) is called the bivariate Poincaré series of the algebra of covariants ğ’žğ‘‘. It is clear that the series 𝒫𝑑(𝑧,0) is the Poincaré series of the algebra ℐ𝑑 and the series 𝒫𝑑(𝑧,1) is the Poincaré series of the algebra ğ’žğ‘‘ 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 𝒫𝑑(𝑧,𝑡) were calculated by Sylvester, see in [1, 2] the big tables of 𝒫𝑑(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 ℐ𝑑. In the paper we have proved a Cayley-Sylvester-type formula for calculating of dim(ğ’žğ‘‘)𝑖,𝑗 and a Springer-type formula for calculation of 𝒫𝑑(𝑧,𝑡). By using the formula, the bivariate Poincaré series 𝒫𝑑(𝑧,𝑡) is calculated for 𝑑≤20.

2. Cayley-Sylvester-Type Formula for dim(ğ’žğ‘‘)𝑖,𝑗

To begin, we give a proof of the Cayley-Sylvester-type formula for the dimension of the graded component (ğ’žğ‘‘)𝑖,𝑗.

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

Let 𝔲2 be the maximal unipotent subalgebra of 𝔰𝔩2. The algebra 𝒮𝑑, defined by 𝒮𝑑𝑉∶=𝑆𝑑𝔲2=𝑉𝑣∈𝑆𝑑∣𝐷1,(𝑣)=0(2.2) is called the algebra of semi-invariants of the binary form of degree 𝑑. For any element 𝑣∈𝒮𝑑, a natural number 𝑠 is called the order of the element 𝑣 if the number 𝑠 is the smallest natural number such that 𝐷𝑠2(𝑣)≠0,𝐷2𝑠+1(𝑣)=0.(2.3) It is clear that any semi-invariant 𝑣∈𝒮𝑑 of order 𝑖 is the highest weight vector for an irreducible 𝔰𝔩2-module of the dimension 𝑖+1 in 𝑆(𝑉𝑑).

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 𝒮𝑑.

The algebra 𝑆(𝑉𝑑) is ℕ-graded 𝑆𝑉𝑑=𝑆0𝑉𝑑+𝑆1𝑉𝑑+⋯+𝑆𝑖𝑉𝑑+⋯,(2.4) and each 𝑆𝑖(𝑉𝑑) is a completely reducible representation of the Lie algebra 𝔰𝔩2. Thus, the following decomposition holds 𝑆𝑖𝑉𝑑≅𝛾𝑑(𝑖,0)𝑉0+𝛾𝑑(𝑖,1)𝑉1+⋯+𝛾𝑑(𝑖,𝑑⋅𝑛)𝑉𝑑⋅𝑖,(∗) here 𝛾𝑑(𝑖,𝑗) is the multiplicity of the representation 𝑉𝑗 in the decomposition of 𝑆𝑖(𝑉𝑑). On the other hand, the multiplicity 𝛾𝑑(𝑖,𝑗) of the representation 𝑉𝑗 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. î€·ğ’ždim𝑑𝑖,𝑗=𝛾𝑑(𝑖,𝑗).(2.5)

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

A formal sum Char(𝑊)=𝑘∈Λ𝑊𝑛𝑊(𝑘)ğ‘žğ‘˜,(2.6) is called the character of a representation 𝑊, here 𝑛𝑊(𝑘) denotes the multiplicity of the weight 𝑘∈Λ𝑊. Since, the multiplicity of any weight of the irreducible representation 𝑉𝑑 is equal to 1, we have 𝑉Char𝑑=ğ‘žâˆ’ğ‘‘+ğ‘žâˆ’ğ‘‘+2+⋯+ğ‘žğ‘‘âˆ’2+ğ‘žğ‘‘.(2.7)

The character Char(𝑆𝑛(𝑉𝑑)) of the representation 𝑆𝑛(𝑉𝑑) equals ğ»ğ‘›î€·ğ‘žâˆ’ğ‘‘,ğ‘žâˆ’ğ‘‘+2,…,ğ‘žğ‘‘î€¸,(2.8) (see [5]), where 𝐻𝑛(𝑥0,𝑥1,…,𝑥𝑑) is the complete symmetrical function 𝐻𝑛𝑥0,𝑥1,…,𝑥𝑑=|𝛼|=𝑛𝑥𝛼00𝑥𝛼11⋯𝑥𝛼𝑑𝑑,|𝛼|=𝑖𝛼𝑖.(2.9)

By replacing 𝑥𝑘 with ğ‘žğ‘‘âˆ’2𝑘, 𝑘=0,…,𝑑, we obtain the specialized expression for Char(𝑆𝑛(𝑉𝑑)): 𝑆Char𝑛𝑉𝑑=|𝛼|=ğ‘›î€·ğ‘žğ‘‘î€¸ğ›¼0î€·ğ‘žğ‘‘âˆ’2⋅1𝛼1â‹¯î€·ğ‘žğ‘‘âˆ’2𝑑𝛼𝑑=|𝛼|=ğ‘›ğ‘žğ‘‘ğ‘›âˆ’2(𝛼1+2𝛼2+⋯+𝑑𝛼𝑑)=𝑑𝑛𝑘=−𝑑𝑛𝜔𝑑𝑛,𝑑𝑛−𝑘2î‚ğ‘žğ‘˜,(2.10) here 𝜔𝑑(𝑛,(𝑑𝑛−𝑘)/2) is the number nonnegative integer solutions of the equation 𝛼1+2𝛼2+⋯+𝑑𝛼𝑑=𝑑𝑛−𝑘2,(2.11) on the assumption that 𝛼0+𝛼1+⋯+𝛼𝑑=𝑛. In particular, the coefficient of ğ‘ž0 (the multiplicity of zero weight ) is equal to 𝜔𝑑(𝑛,𝑑𝑛/2), and the coefficient of ğ‘ž1 is equal to 𝜔𝑑(𝑛,(𝑑𝑛−1)/2).

On the other hand, the decomposition (*) implies the equality for the characters: 𝑆Char𝑛𝑉𝑑=𝛾𝑑𝑉(𝑛,0)Char0+𝛾𝑑𝑉(𝑛,1)Char1+⋯+𝛾𝑑𝑉(𝑛,𝑑𝑛)Char𝑑𝑛.(2.12) We can summarize what we have shown so far in the following.

Theorem 2.2. î€·ğ’ždim𝑑𝑖,𝑗=𝜔𝑑𝑖,𝑑𝑖−𝑗2−𝜔𝑑𝑖,𝑑𝑖−(𝑗+2)2.(2.13)

Proof. The weight 𝑗 appears once in any representation 𝑉𝑘, for 𝑘=𝑗mod2,𝑘≥𝑗. Therefore 𝜔𝑑𝑖,𝑑𝑖−𝑗2=𝛾𝑑(𝑖,𝑗)+𝛾𝑑(𝑖,𝑗+2)+⋯+𝛾𝑑(𝑖,𝑗+4)+⋯.(2.14) Similarly, 𝜔𝑑𝑖,𝑑𝑖−(𝑗+2)2=𝛾𝑑(𝑖,𝑗+2)+𝛾𝑑(𝑖,𝑗+4)+⋯+𝛾𝑑(𝑖,𝑗+6)+⋯.(2.15) Thus, 𝜔𝑑𝑖,𝑑𝑖−𝑗2−𝜔𝑑𝑖,𝑑𝑖−(𝑗+2)2=𝛾𝑑(𝑖,𝑗).(2.16) By using Lemma 2.1 we obtain î€·ğ’ždim𝑑𝑖,𝑗=𝜔𝑑𝑖,𝑑𝑖−𝑗2−𝜔𝑑𝑖,𝑑𝑖−(𝑗+2)2.(2.17)

For another proof of the formula see [3].

Note that the original Cayley-Sylvester formula is ℐdim𝑑𝑛=𝜔𝑑𝑛,𝑑𝑛2−𝜔𝑑𝑛,𝑑𝑛2.−1(2.18) Also, in [6] we proved that î€·ğ’ždim𝑑𝑛=𝜔𝑑𝑛,𝑑𝑛2+𝜔𝑑𝑛,𝑑𝑛−12.(2.19) Here (ℐ𝑑)𝑛,(ğ’žğ‘‘)𝑛 are the components of standard grading of the algebras ℐ𝑑, ğ’žğ‘‘ under degree.

3. Calculation of dim(𝐶𝑑)𝑖,𝑗

It is well known that the number 𝜔𝑑(𝑖,(𝑑𝑖−𝑗)/2) of nonnegative integer solutions of the following system 𝛼1+2𝛼2+⋯+𝑑𝛼𝑑=𝑑𝑖−𝑗2,𝛼0+𝛼1+⋯+𝛼𝑑=𝑖,(3.1) is given by the coefficient of 𝑧𝑛𝑡(𝑑𝑖−𝑗)/2 of the generating function𝑓𝑑1(𝑧,𝑡)=(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡𝑑.(3.2) We will use the notation [𝑥𝑘]𝐹(𝑥) to denote the coefficient of 𝑥𝑘 in the series expansion of 𝐹(𝑥)∈ℂ[[𝑥]]. Thus 𝜔𝑑𝑖,𝑑𝑖−𝑗2=𝑧𝑖𝑡(𝑑𝑖−𝑗)/2𝑓𝑑(𝑧,𝑡).(3.3) It is clear that 𝜔𝑑𝑖,𝑑𝑖−𝑗2=𝑧𝑖𝑡𝑑𝑖−𝑗𝑓𝑑𝑧,𝑡2=𝑧𝑡𝑑𝑖𝑡𝑗𝑓𝑑𝑧,𝑡2.(3.4)

Similarly, the number 𝜔𝑑(𝑖,(𝑑𝑖−(𝑗+2))/2) of nonnegative integer solutions of the following system𝛼1+2𝛼2+⋯+𝑑𝛼𝑑=𝑑𝑖−(𝑗+2)2,𝛼0+𝛼1+⋯+𝛼𝑑=𝑖,(3.5)

equals 𝑧𝑖𝑡(𝑑𝑖−(𝑗+2))/2𝑓𝑑𝑧(𝑧,𝑡)=𝑖𝑡𝑑𝑖−(𝑗+2)𝑓𝑑𝑧,𝑡2=𝑧𝑡𝑑𝑖𝑡𝑗+2𝑓𝑑𝑧,𝑡2.(3.6)

Therefore, 𝜔𝑑𝑖,𝑑𝑖−𝑗2−𝜔𝑑𝑖,𝑑𝑖−(𝑗+2)2=𝑧𝑡𝑑𝑖𝑡𝑗𝑓𝑑𝑧,𝑡2−𝑧𝑡𝑑𝑖𝑡𝑗+2𝑓𝑑𝑧,𝑡2=𝑧𝑡𝑑𝑖𝑡𝑗−𝑡𝑗+2𝑓𝑑𝑧,𝑡2=𝑧𝑡𝑑𝑖𝑡𝑗1−𝑡2𝑓𝑑𝑧,𝑡2=𝑧𝑖𝑡𝑑𝑖−𝑗1−𝑡2𝑓𝑑𝑧,𝑡2.(3.7) Thus, the following statement holds.

Theorem 3.1. The number dim(ğ’žğ‘‘)𝑖,𝑗 of linearly independent covariants of degree 𝑖 and order 𝑗 for the binary d- form is given by the formula 𝐶dim𝑑𝑖,𝑗=𝑧𝑖𝑡𝑑𝑖−𝑗1−𝑡2(1−𝑧)1−𝑧𝑡2⋯1−𝑧𝑡2𝑑.(3.8)
It is clear that 𝑧𝑖𝑡𝑑𝑖−𝑗1−𝑡2(1−𝑧)1−𝑧𝑡2⋯1−𝑧𝑡2𝑑=𝑧𝑖𝑡(𝑑𝑖−𝑗)/21−𝑡(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡𝑑.(3.9) By using the decomposition 1(1−𝑧)(1−𝑧𝑡)⋯1−𝑧𝑡𝑑=âˆžî“ğ‘˜=0𝑑𝑖𝑡𝑧𝑖,(3.10) where î€ºğ‘‘ğ‘›î€»ğ‘ž is the ğ‘ž-binomial coefficient îƒ¬ğ‘‘ğ‘›îƒ­ğ‘žî€·âˆ¶=1âˆ’ğ‘žğ‘‘+11âˆ’ğ‘žğ‘‘+2⋯1âˆ’ğ‘žğ‘‘+𝑛(1âˆ’ğ‘ž)1âˆ’ğ‘ž2⋯(1âˆ’ğ‘žğ‘›),(3.11) one obtains the well-known formula 𝐶dim𝑑𝑖,𝑗=𝑡(𝑑𝑖−𝑗)/2𝑑𝑖(1−𝑡)𝑡,(3.12) for instance, see [3].

4. Explicit Formula for 𝒫𝑑(𝑧,𝑡)

Let us prove Springer-type formula for the bivariate Poincaré series 𝒫𝑑(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 Ψ𝑑,∶ℤ[[𝑧,𝑡]]⟶ℤ[[𝑧,𝑡]](4.1) in the following way Ψ𝑑𝑧𝑖𝑡𝑗=𝑧𝑖𝑡𝑑𝑖−𝑗,if𝑑𝑖−𝑗≥0,0,if𝑑𝑖−𝑗<0.(4.2)

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

Lemma 4.1. 𝒫𝑑(𝑧,𝑡)=Ψ𝑑1−𝑡2(1−𝑧𝑡)1−𝑧𝑡2⋯1−𝑧𝑡2𝑑.(4.3)

Proof. Theorem 2.2 implies that dim(𝐶𝑑)𝑖,𝑗=[𝑧𝑖𝑡𝑑𝑖−𝑗]𝑓𝑑(𝑧,𝑡2). Then 𝒫𝑑(𝑧,𝑡)=âˆžî“ğ‘–,𝑗=0𝐶dim𝑑𝑖,𝑗𝑧𝑖𝑡𝑗=âˆžî“ğ‘–,𝑗=0𝑧𝑖𝑡𝑑𝑖−𝑗𝑓𝑑𝑧,𝑡2𝑧𝑖𝑡𝑗=Ψ𝑑𝑓𝑑𝑧,𝑡2.(4.4)

Let 𝜓𝑛∶ℤ[[𝑡]]→ℤ[[𝑡,𝑧]],𝑛∈ℤ be a ℂ-linear function defined by 𝜓𝑛(𝑡𝑚)∶=𝑧𝑖𝑡𝑗𝑘,where𝑖∶=minî…žâˆ£ğ‘›ğ‘˜î…žî€¾âˆ’ğ‘šâ‰¥0,𝑗=𝑛𝑖−𝑚,(4.5)

for 𝑖,𝑗,𝑚,𝑛∈ℕ. Note that 𝜓𝑛(𝑡0)=1 and 𝜓1(𝑡𝑚)=𝑧𝑚, 𝜓0(𝑡𝑚)=1. Also, put 𝜓𝑛(𝑡𝑚)=0 for 𝑛<0. It is clear that 𝜓𝑛(𝑡𝑛𝑖−𝑗)=𝑧𝑖𝑡𝑗 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 𝜓𝑛(𝑅(𝑡𝑛)𝐻(𝑡))=𝑅(𝑧)𝜓𝑛(𝐻(𝑡)).
(ii) For 𝑅(𝑡)∈ℂ[[𝑡]] and for 𝑛,𝑘∈ℕ holds Ψ𝑛𝑅(𝑡)1−𝑧𝑡𝑘=âŽ§âŽªâŽ¨âŽªâŽ©ğœ“ğ‘›âˆ’ğ‘˜(𝑅(𝑡))1−𝑧𝑡𝑛−𝑘,𝑛≥𝑘,0,if𝑛<𝑘.(4.6)

Proof. (i) The statement follows from the linearity of the function 𝜓𝑛 and from the following simple observation: 𝜓𝑛𝑡𝑛𝑘𝑡𝑛𝑖−𝑗=𝜓𝑛𝑡𝑛(𝑘+𝑖)−𝑗=𝑧𝑘+𝑖𝑡𝑗=𝑧𝑘𝑧𝑖𝑡𝑗=𝑧𝑘𝜓𝑛𝑡𝑛𝑖−𝑗,(4.7) for 𝑛𝑖−𝑗≥0 and 𝑗<𝑛.
(ii) Let ∑𝑅(𝑡)=âˆžğ‘š=0ğ‘Žğ‘šğ‘¡ğ‘š. Then for 𝑘<𝑛 we have Ψ𝑛𝑅(𝑡)1−𝑧𝑡𝑘=Î¨ğ‘›îƒ©âˆžî“ğ‘š,𝑠=0ğ‘Žğ‘šğ‘¡ğ‘šî€·ğ‘§ğ‘¡ğ‘˜î€¸ğ‘ îƒª=Î¨ğ‘›îƒ©âˆžî“ğ‘š,𝑠=0ğ‘Žğ‘šğ‘§ğ‘ ğ‘¡ğ‘˜ğ‘ +𝑚=(𝑛−𝑘)𝑠−𝑚≥0ğ‘Žğ‘šğ‘§ğ‘ ğ‘¡(𝑛−𝑘)𝑠−𝑚=âˆžî“ğ‘š,𝑠=0ğ‘Žğ‘šğœ“ğ‘›âˆ’ğ‘˜(𝑡𝑚)𝑧𝑡𝑛−𝑘𝑠=âˆžî“ğ‘š=0ğ‘Žğ‘šğœ“ğ‘›âˆ’ğ‘˜(𝑡𝑚)11−𝑧𝑡𝑛−𝑘=𝜓𝑛−𝑘(𝑅(𝑡))1−𝑧𝑡𝑛−𝑘.(4.8)

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

Theorem 4.3. 𝒫𝑑(𝑧,𝑡)=0≤𝑘<𝑑/2𝜓𝑑−2𝑘(−1)𝑘𝑡𝑘(𝑘+1)1−𝑡2𝑡2,𝑡2𝑘𝑡2,𝑡2𝑑−𝑘11−𝑧𝑡𝑑−2𝑘,(4.9) here (ğ‘Ž,ğ‘ž)𝑛=(1âˆ’ğ‘Ž)(1âˆ’ğ‘Žğ‘ž)⋯(1âˆ’ğ‘Žğ‘žğ‘›âˆ’1) is ğ‘ž-shifted factorial.

Proof. Consider the partial fraction decomposition of the rational function 𝑓𝑑(𝑧,𝑡2): 𝑓𝑑𝑧,𝑡2=𝑑𝑘=0𝑅𝑘(𝑧)1−𝑡𝑧2𝑘.(4.10) It is easy to see, that 𝑅𝑘(𝑡)=lim𝑧→𝑡−2𝑘𝑓𝑑𝑧,𝑡21−𝑧𝑡2𝑘=lim𝑧→𝑡−2𝑘1−𝑡2(𝑧,𝑡)𝑑+11−𝑧𝑡2𝑘=1−𝑡21−𝑡−2𝑘1−𝑡2−2𝑘⋯1−𝑡2(𝑘−1)−2𝑘1−𝑡2(𝑘+1)−2𝑘⋯1−𝑡2𝑑−2𝑘=𝑡2𝑘+(2𝑘−2)+⋯+21−𝑡2𝑡2𝑘𝑡−12𝑘−2⋯𝑡−12−11−𝑡2⋯1−𝑡2𝑑−2𝑘=(−1)𝑘𝑡𝑘(𝑘+1)1−𝑡2𝑡2,𝑡2𝑘𝑡2,𝑡2𝑑−𝑘.(4.11) Using the above Lemmas we obtain 𝒫𝑑(𝑧,𝑡)=Ψ𝑑𝑓𝑑𝑧,𝑡2=Ψ𝑑𝑛𝑘=0𝑅𝑘𝑡21−𝑧𝑡2𝑘=0≤𝑘<𝑑/2𝜑𝑑−2𝑘(−1)𝑘𝑡𝑘(𝑘+1)1−𝑡2𝑡2,𝑡2𝑘𝑡2,𝑡2𝑑−𝑘11−𝑧𝑡𝑑−2𝑘.(4.12)

Corollary 4.4. A denominator of the bivariate Poincaré series 𝒫𝑑(𝑧,𝑡), 𝑑>2 can be written in the form [](𝑑+1)/2𝑘=01−𝑧𝑡𝑑−2𝑘𝑑−2𝑖=11−𝑧𝑘𝑖,(4.13) where 𝑘1,𝑘2,…,𝑘𝑑−2 are the degrees of elements of homogeneous system of parameters for the algebra of invariants ℐ𝑑.

Proof. The formula of Theorem 4.3 implies that the bivariate Poincaré series has the form 𝒫𝑑𝑃(𝑧,𝑡)=𝑑(𝑧,𝑡)∏[(𝑑+1)/2]𝑘=01−𝑧𝑡𝑑−2𝑘𝑅𝑑,(𝑧)(4.14) for some polynomials 𝑃𝑑(𝑧,𝑡),𝑅𝑑(𝑧). Thus, the Poincaré series for the algebra of invariants ℐ𝑑 has the form 𝒫𝑑𝑃(𝑧,0)=𝑑(𝑧,0)𝑅𝑑.(𝑧)(4.15) The algebra of invariants ℐ𝑑 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 𝑅𝑑(𝑧) of its Poincaré series can be written in the following way 𝑅𝑑(𝑧)=(1−𝑧𝑘1)(1−𝑧𝑘2)⋯(1−𝑧𝑘𝑑−2), where 𝑘1,𝑘2,…,𝑘𝑑−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 𝜓𝑛𝑅(𝑡)1−𝑡𝑘11−𝑡𝑘2⋯1−𝑡𝑘𝑚=𝜓𝑛𝑅(𝑧)𝑄𝑛𝑡𝑘1𝑄𝑛𝑡𝑘2𝑄𝑛𝑡𝑘𝑚1−𝑧𝑘11−𝑧𝑘2⋯1−𝑧𝑘𝑚,(5.1) here 𝑄𝑛(𝑡)=1+𝑡+𝑡2+⋯+𝑡𝑛−1, and 𝑘𝑖 are natural numbers.

Proof. Taking into account Lemma 4.2 we get 𝜓𝑛𝑔(𝑡)1−𝑡𝑚=𝜓𝑛𝑔(𝑡)1−𝑡𝑛𝑚1−𝑡𝑛𝑚1−𝑡𝑚=11−𝑡𝑚𝜓𝑛𝑔(𝑡)1−𝑡𝑛𝑚1−𝑡𝑚=11−𝑡𝑚𝜓𝑛𝑔(𝑡)1+𝑡𝑚+(𝑡𝑚)2+⋯+(𝑡𝑚)𝑛−1=11−𝑡𝑚𝜓𝑛𝑔(𝑡)𝑄𝑛(𝑡𝑚).(5.2) In a similar fashion we prove the general case.

By using Lemma 5.1 the bivariate Poincaré series 𝒫𝑑(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:𝒫11(𝑧,𝑡)=1−𝑧𝑡,𝒫21(𝑧,𝑡)=1−𝑧𝑡21−𝑧2,𝒫3(𝑧𝑧,𝑡)∶=2𝑡2−𝑧𝑡+1(1−𝑧𝑡)1−𝑧𝑡31−𝑧4,𝒫4𝑧(𝑧,𝑡)=2𝑡4−𝑧𝑡2+11−𝑧𝑡21−𝑡4𝑧1−𝑧21−𝑧3,𝒫5𝑝(𝑧,𝑡)=5(𝑧,𝑡)(1−𝑧𝑡)1−𝑡3𝑧1−𝑧𝑡51−𝑧81−𝑧61−𝑧4,𝑝5(𝑧,𝑡)=1+𝑧7𝑡3−𝑧6𝑡4+𝑧2𝑡2+2𝑧7𝑡−𝑧5𝑡5−𝑧8𝑡2−2𝑧8𝑡6−𝑧8𝑡4+𝑧5𝑡3+𝑧5𝑡+𝑧9𝑡7−𝑧10𝑡6+𝑧10𝑡2−𝑧10𝑡4−𝑧11𝑡3+𝑧9𝑡3−𝑡3𝑧−𝑧6+𝑧4𝑡4−𝑧𝑡+𝑧2𝑡6+𝑧2𝑡4+𝑧12+𝑧14𝑡6−𝑧13𝑡−𝑧13𝑡5−𝑧13𝑡3−𝑧15𝑡7+𝑧14𝑡4−𝑧3𝑡7+𝑧7𝑡5.(5.3)