My research interests lie in the fields of applied algebraic geometry, commutative algebra and algebraic transformation groups theory.
In particular, currently I am interested in Gröbner and Khovanskii bases, their applications to the polynomial systems solving and related topics such as algorithms in computational algebraic geometry and their implementation.
Khovanskii bases for semimixed systems of polynomial equations – a case of approximating stationary nonlinear Newtonian dynamicsWith Paul Breiding, Javier del Pino, Mateusz Michałek and Oded Zilberberg
We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory of Khovanskii bases, combined with toric geometry, the Bernstein-Khovanskii-Kushnirenko (BKK) Theorem, and fiber products.
As a direct application of this theory, we solve the problem of counting the number of approximate stationary states for coupled driven nonlinear resonators. We use our main theorems, that is, the generalized BKK Theorem and the Decoupling Theorem, to count the number of (complex) solutions of the polynomial system for an arbitrary degree of nonlinearity and any number of resonators.
A short proof for the parameter continuation theoremWith Paul Breiding
The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.
On orbits of automorphism groups on horospherical varietiesWith Sergey Gaifullin and Anton Shafarevich
In this paper we describe orbits of automorphism group on a horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (maybe non-normal) toric varieties a description of orbits of automorphism group in terms of corresponding weight monoid is obtained.
Commutative actions on smooth projective quadricsWith Sergey Gaifullin and Anton Trushin
By a commutative action on a smooth quadric Q_n we mean an effective action of a commutative connected algebraic group on Q_n with an open orbit. We show that for n≥3 all commutative actions on Q_n are additive actions described by Sharoiko in 2009. So there is a unique commutative action on Q_n up to equivalence. For n=2 there are three commutative actions on Q_2 up to equivalence, for n=1 there are two commutative actions on Q_1 up to equivalence.