arXiv subjects

S. Yu. Orevkov

Publications and source records attributed to S. Yu. Orevkov.

24 records · Page 2Linked to original sources

arXiv2013

On modular computation of Groebner bases with integer coefficients

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gröbner base of $I$ under the assumption that the Gröbner bases of the ideal $\Bbb Q I$ of the ring $\Bbb Q[X]$ and the the ideals $I\otimes(\Bbb Z/m\Bbb Z)$ of the rings $(\Bbb Z/m\Bbb Z)[X]$ are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.

math.AC↗

arXiv2013

Cubic Hecke Algebras and Invariants of Transversal Links

We propose a purely algebraic approach to construct invariants of transversal links in the standard contact structure on the 3-sphere generalizing Jones' approach to invariant of usual links. The only geometry used is the analogue of Alexander and Markov theorems. More precisely, we construct a trace on a certain cubic Hecke algebra which is invariant under positive Markov moves only (we propose to call it a semi-Markov trace). The trace takes its values in the quotient of a polynomial ring by a certain ideal. An algorithm for computing a Groebner base of the ideal is given.

math.GT↗

arXiv2008

Agnihotri-Woodward-Belkale polytope and the intersection of Klyachko cones

Agnihotri-Woodward-Belkale polytope $Δ$ (resp. Klyachko cone $K$) is the set of solutions of the multiplicative (resp. additive) Horn's problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) $n\times n$ matrices satisfying $AB=C$ (resp. $A+B=C$). $K$ is the tangent cone of $Δ$ at the origin. The group $G=\Bbb Z_n \oplus \Bbb Z_n$ acts naturally on $Δ$. In this note, we report on a computer calculation which shows that $Δ$ coincides with the intersection of $gK$, $g\in G$, for $n\le 14$ but does not coincide for $n=15$. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU(n).

math.CO↗

arXiv2005

Algebraic curve in the unit ball in C^2 passing through the center, all whose boundary components are arbitrarily short

We prove that curves indicated in the title exist. This results answers to a question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal number of boundary components of a curve in the unit ball passing through the center, under the condition that all these components are shorter than a given number. More precisely, we discuss the order of growth of the number of the components as their maximal length tends to zero.

math.CV↗

arXiv2003

Markov theorem for transversal links

It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.

math.GT↗

arXiv2000

Asymptotic growth of the number of classes of real plane algebraic curves when the degree increases

The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are studied. It is shown, in particular, that $log I_d\asypt d^2$. Other related problems and their higher dimensional generalisations are discussed.

math.AG↗