关于8次二元形式的完备协变系

The degree of the algebra of covariants of a binary form

We calculate the degree of the algebra of covariants $\mathcal{C}_d$ for binary $d$-form. Also, for the degree we obtain its integral representation and asymptotic behavior.

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 $d_1,d_2 \leq 20.$

关于二元五次曲线的Hermite不变量

On Hermite's invariant for binary quintics

Hermite不变量H是二元五次对合超曲面的定义方程。本文分析了H.我们确定了这个超曲面的奇异轨迹，并证明它是五次线性协变的完全交集。这个超曲面的投射对偶可以通过对合来与它自身等同。证明了H的Jacobi理想是高度为2的完全理想，并刻画了它的SL_2-等变最小分解。最后一节给出了二元形式协变的向矢的一般形式，并用它计算了H的向矢。

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 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…

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。

关于7次二元形式的完全不变量系

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

计算了7次二元形式的不变代数的齐次生成元的极小系。