Hilbert polynomials for the algebra of invariants of binary d-form

Анотація

We consider one of the fundamental problems of classical invariant theory - the research  of Hilbert polynomials for the algebra of invariants  of the Lie group $S{L}_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. It is well-known that the Hilbert function of the algebra $S{L}_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form ${\mathcal{C}}_d=\mathrm{ℂ}{\left[V_d\oplus {\mathrm{ℂ}}^2\right]}^{S{L}_2}$ (here $V_d$ is the  $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was  generalized to  algebra of  invariants of  binary $d$-form ${\mathcal{I}}_d=\mathrm{ℂ}{\left[V_d\right]}^{S{L}_2}$. But the Cayley-Sylvester formula is  not  expressed in terms of polynomials. In our article we consider the problem of computing  the Hilbert polynomials for the algebra  of invariants of  binary $d$-form.