Multivariate Poincar\'e series for algebras of $SL_2$-invariants

TL;DR

This paper derives formulas for the multivariate Poincaré series of algebras of joint covariants and invariants of multiple binary forms, providing tools for their explicit computation.

Contribution

It introduces explicit formulas for computing the multivariate Poincaré series of algebras of $SL_2$-invariants and covariants of several binary forms.

Findings

Formulas for multivariate Poincaré series are established.

Explicit computational methods are provided.

Results facilitate analysis of algebraic invariants.

Abstract

Let $\mathcal{C}_{\mathbi{d}},$ $\mathcal{I}_{\mathbi{d}},$ $\mathbi{d}{=}(d_1,d_2,..., d_n)$ be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,..., d_n.$ Formulas for computation of the multivariate Poincar\'e series $\mathcal{P}(\mathcal{C}_{\mathbi{d}},z_1,z_2,...,z_n,t)$ and $\mathcal{P}(\mathcal{I}_{\mathbi{d}},z_1,z_2,...,z_n)$ are found.