arXiv subjects

Andrei Minchenko

Publications and source records attributed to Andrei Minchenko.

7 recordsLinked to original sources

arXiv2018

Degree bound for toric envelope of a linear algebraic group

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group $G \subset \mathrm{GL}_n(C)$ can be arbitrarily large even for $n = 1$. One of the key ingredients of Hrushovski's algorithm for computing the Galois group of a linear differential equation was an idea to `approximate' every algebraic subgroup of $\mathrm{GL}_n(C)$ by a `similar' group so that the degree of the latter is bounded uniformly in $n$. Making this uniform bound computationally feasible is crucial for making the algorithm practical. In this paper, we derive a single-exponential degree bound for such an approximation (we call it toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound for the first step of the Hrushovski's algorithm due to Feng to a single-exponential bound. For the cases $n = 2, 3$ often arising in practice, we further refine our general bound.

math.AG↗

arXiv2017

Triviality of differential Galois cohomologies of linear differential algebraic groups

We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear differential equation with parameters.

math.AG↗

arXiv2017

Etale representations for reductive algebraic groups with factors $Sp_n$ or $SO_n$

A complex vector space $V$ is an \'etale $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit and $\dim G=\dim V$. Such a module is called super-\'etale if the stabilizer of a point in the open orbit is trivial. Popov proved that reductive algebraic groups admitting super-\'etale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-\'etale module is always isomorphic to a product of general linear groups. In light of previously available examples, one can conjecture more generally that in such a group all simple factors are either $SL_n$ for some $n$ or $Sp_2$. We show that this is not the case by constructing a family of super-\'etale modules for groups with a factor $Sp_n$ for arbitrary $n\geq1$. A similar construction provides a family of \'etale modules for groups with a factor $SO_n$, which shows that groups with \'etale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples of \'etale and super-\'etale modules for reductive groups. Finally, we show that the exceptional groups $F_4$ and $E_8$ cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a linear \'etale representation.

math.RT↗

arXiv2016

Calculating Galois groups of third order linear differential equations with parameters

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

math.RT↗

arXiv2015

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

math.AC↗

arXiv2014

On central extensions of simple differential algebraic groups

We consider central extensions $Z\hookrightarrow E\twoheadrightarrow G$ in the category of linear differential algebraic groups. We show that if $G$ is simple non-commutative and $Z$ is unipotent with the differential type smaller than that of $G$, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.

math.RT↗

arXiv2010

Enhanced Dynkin diagrams and Weyl orbits

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.

math.RT↗