The Poincar\'e series of the joint invariants and covariants of the two binary forms

TL;DR

This paper derives formulas for the Poincaré series of the algebras of joint invariants and covariants of two binary forms and computes these series for degrees up to 20, advancing understanding of their algebraic structure.

Contribution

It provides explicit formulas for the Poincaré series of joint invariants and covariants, enabling computations for degrees up to 20, which was previously not systematically done.

Findings

Formulas for Poincaré series are derived.

Series computed for degrees up to 20.

Enhanced understanding of algebraic structures of joint invariants and covariants.

Abstract

Let $\mathcal{I}_{d_1,d_2}$ and $\mathcal{C}_{d_1,d_2}$ be the algebras of joint invariants and joint covariants of the two binary forms of degrees $d_1$ and $d_2.$ Formulas for computation of the Poincar\'e series $\mathcal{PI}_{d_1,d_2}(z),$ $ \mathcal{PC}_{d_1,d_2}(z)$ of the algebras is found. By using these formulas, we have computed the series for $d_1,d_2 \leq 20.$