arXiv subjects
Publications and source records attributed to Leonid Bedratyuk.
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 $d_1,d_2 \leq 20.$
A formula for computation of the Poincaré series $P_d(z)$ of the algebra of the covariants of binary $d$-form is found. By using it, we have computed the $P_d(z)$ for $d \leq 20.$
Analogue of Springer's formula for the Poincaré series of the algebra invariants of ternary form is found.
The number $ν_{n,d}(k)$ of linearly independed homogeneous invariants of degree $k$ for the $n$-ary form of degree $d$ is calculated.
The number $ν_d(n)$ of linearly independed homogeneous invariants of degree $n$ for the ternary form of degree $d$ is calculated.
Let $L$ be one of the finite dimensional Lie algebras $W_n({\bf m}),$ $S_n({\bf m}),$ $ H_n({\bf m})$ of Cartan type over an algebraically closed field of prime characteristic $p>0.$ For an elements $F$ of the symmetrical algebra $S(L)$ we found necessary and sufficient condition in order to the element $ad(\partial_1)^{p^{m_1}-1} ad(\partial_2)^{p^{m_2}-1}... ad(\partial_n)^{p^{m_n}-1}(F)$ belongs to the symmetrical invariants algebra $S(L)^L.$ Also, for $p=3,5$ the algebra of symmetrical invariants $S(H_2)^{H_2}$ is calculated in explicit way.
Analogues of invariant theory's well-known Roberts theorem are proved for ternary forms. We established that covariants, contravariants and mixed concomitants of a ternary form are uniquely determined by their lead coefficients.
A minimal system of 147 homogeneous generating elements of the algebra of covariants for the binary form of degree 7 is calculated.
A minimal system of homogeneous generating elements of the algebra of covariants for the binary form of degree 8 is calculated.
A minimal system of homogeneous generating elements of the invariants algebra for the binary form of degree 7 is calculated.
An explicit form of first order PDE for invariants of binary form are found. By solving the equation a minimal generation set for a ring of invariants and theirs syzygies are calculated in the cases $n\leq 6$ and $n=8.$