International Scholarly Research Network ISRN Algebra Volume 2011, Article ID 312789, 11 pages doi:10.5402/2011/312789 Research Article Bivariate Poincar´ e Series for the Algebra of Covariants of a Binary Form Leonid Bedratyuk Khmelnytsky National University, Instituts’Ka Street 11, 29016 Khmelnytsky, Ukraine Correspondence should be addressed to Leonid Bedratyuk, leonid.uk@gmail.com Received 25 April 2011; Accepted 17 May 2011 Academic Editor: E. Giuli Copyright q 2011 Leonid Bedratyuk. 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. A formula for computation of the bivariate Poincar´ e 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, . Consider the corresponding action of the group G on the coordinate rings V d  and V d ⊕ 2 . Denote by I d  V d  G and by C d  V d ⊕ 2  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, t, P d z, t ∞  i,j 0 dim  C d  i,j  z i t j , 1.2