2019
Том 71
№ 8

All Issues

Bedratyuk L. P.

Articles: 4
Article (Ukrainian)

Derivations and Identities for Kravchuk Polynomials

Bedratyuk L. P.

↓ Abstract   |   Full text (.pdf)

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

Bedratyuk L. P.

↓ Abstract   |   Full text (.pdf)

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

Bedratyuk L. P.

↓ Abstract   |   Full text (.pdf)

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

Bedratyuk L. P.

↓ Abstract   |   Full text (.pdf)

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 $n ≤ 6$.