Victor Ufnarovski

Non-Commutative Gröbner Bases and Anick’s Resolution

Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Grobner bases, Hilbert series and Anick’s resolution are formulated. Special attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are investigated. The existence of finite state automata, defining the behavior of the Hilbert series is discussed.

The Computer Algebra Package Bergman: Current State

Bergman is a special-purpose system for computations in commutative and purely non-commutative graded algebra.

On Sagbi Bases and Resultants

A resultant-type identity for univariate polynomials is proved and applied to characterization of SAGBI bases of subalgebras, generated by two polynomials. Besides a new condition for polynomials f(x) and g(x) to form a SAGBI basis, expressed in terms of field extensions is derived.

Using resultants for SAGBI basis verification in the univariate polynomial ring

A resultant-type identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived.