On complete system of covariants for the binary form of degree 7

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.

Covariants of binary forms and new identities for Bernoulli, Euler and Hermite polynomials

Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.

The Poincaré series of the joint invariants and covariants of the two binary forms

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 Poincaré 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 $d_1,d_2 \leq 20.$

The MAPLE package for calculating Poincaré series

We offer a Maple package {\tt Poincare\_Series} for calculating the Poincaré series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of Weitzenböck derivations and for the multivariate Poincaré series of algebras of joint invariants/covariants of several binary forms.

On Hermite's invariant for binary quintics

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal resolution. The last section develops a general formalism for evectants of covariants of binary forms,…

On Hilbert covariants

Let F denote a binary form of order d over the complex numbers. If r is a divisor of d, then the Hilbert covariant H_{r,d}(F) vanishes exactly when F is the perfect power of an order r form. In geometric terms, the coefficients of H give defining equations for the image variety X of an embedding P^r->P^d. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on X. We prove that the ideal generated by the coefficients of H defines X as a scheme. Finally, we exhibi…

On complete system of invariants for the binary form of degree 7

A minimal system of homogeneous generating elements of the invariants algebra for the binary form of degree 7 is calculated.

外尔不变的\(E_8\)雅可比形式环

The ring of Weyl invariant $E_8$ Jacobi forms

研究外尔不变的\(E_8\)弱雅可比形式环，证明其与二元六次式和二元四次式联合协变量环同构且有限生成，借助已知协变量环生成元获极小生成元基。