2019
Том 71
№ 8

# Bedratyuk L. P.

Articles: 4
Article (Ukrainian)

### Derivations and Identities for Kravchuk Polynomials

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1587–1603

We introduce the notion of Kravchuk derivations of the polynomial algebra. It is proved that any element of the kernel of a derivation of this kind gives a polynomial identity satisfied by the Kravchuk polynomials. In addition, we determine the explicit form of isomorphisms mapping the kernel of the basicWeitzenb¨ock derivation onto the kernels of Kravchuk derivations.

Article (Ukrainian)

### Poincare series of the multigraded algebras of SL 2-invariants

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 755-763

Formulas for computation of the multivariate Poincare series $\mathcal{P}(\mathcal{C}_{d}, z_1, z_2,..., z_n,t)$ and $\mathcal{P}(\mathcal{I}_{d}, z_1, z_2,..., z_n)$, are found, where $\mathcal{C}_{d}, \mathcal{I}_{d}, \;\; {d} = (d_1, d_2,..., d_n)$ are multigraded algebras of joint covariants and joint invariants for n binary forms of degrees $d_1, d_2,..., d_n$.

Brief Communications (Ukrainian)

### Analog of the Cayley–Sylvester formula and the Poincaré series for an algebra of invariants of ternary form

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1561–1570

An explicit formula is obtained for the number $ν_d(n)$ of linearly independent homogeneous invariants of degree $n$ of a ternary form of order $d$. A formula for the Poincaré series of the algebra of invariants of the ternary form is also deduced.

Article (Ukrainian)

### Kernels of derivations of polynomial rings and Casimir elements

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 435–452

We propose an algorithm for the evaluation of elements of the kernel of an arbitrary derivation of a polynomial ring. The algorithm is based on an analog of the well-known Casimir element of a finite-dimensional Lie algebra. By using this algorithm, we compute the kernels of Weitzenböck derivation $d(x_i ) = x_{i−1},\; d(x_0) = 0,\;i = 0,…, n$, for the cases where $n ≤ 6$.