A. I. Zobnin

Generalization of the F5 algorithm for calculating Gröbner bases for polynomial ideals

This survey paper presents general approach to the well-known F5 algorithm for calculating Gröbner bases, which was created by Faugère in 2002.

Admissible orderings and finiteness criteria for differential standard bases

We prove that any admissible ordering on ordinary differential monomials in one differential indeterminate can be specified by a canonical set of matrices. The relations between some classes of these orderings are studied. We give criteria of finiteness of differential standard bases and propose an algorithm that computes such bases if they are finite.

Membership problem for differential ideals generated by a composition of polynomials

The question of whether a polynomial belongs to a finitely generated differential ideal remains open. This problem is solved only in some particular cases. In the paper, we propose a method, which reduces the test of membership for fractional ideals generated by a composition of differential polynomials to another, simpler, membership problem.

Algorithm for checking triviality of mixed ideals in the ring of differential polynomials

An algorithm for checking triviality of the ideal [f] + (h1, …, ht) in the ordinary ring of differential polynomials under an additional condition on the polynomial f is suggested. This problem is closely related, on the one hand, to the Kolchin problem on exponents of differential ideals and, on the other hand, to finiteness of differential standard bases.

Generalized Reduction in Rings of Differential Polynomials

In this paper, concepts of algebraic reduction and pseudoreduction in rings of differential polynomials are generalized. Specific features of this generalization, such as the uniqueness of separants and termination of the generalized reduction process, are considered. A method for the construction of generalized almost triangular simplificators similar to that for the construction of medians by Gröbner bases is suggested. A particular case, the generalized reduction in an ordinary polynomial ring, is considered.