arXiv subjects

Viktor Levandovskyy

Publications and source records attributed to Viktor Levandovskyy.

At least 19 recordsLinked to original sources

arXiv2026

An algorithm for annihilator and Bernstein-Sato polynomial of a rational function

The singularity theory of rational functions, i.e., the quotient of two polynomials, has been investigated in the past two decades. The Bernstein-Sato polynomial of a rational function has recently been introduced by Takeuchi. However, only trivial examples are known. We provide an algorithm for computing the Bernstein-Sato polynomial in this context. The strategy is to compute the annihilator of the rational function by using the annihilator of the pair consisting of the numerator and denominator of the quotient. In a natural way a non-vanishing condition on the Bernstein-Sato ideal of the pair appears. This method has been implemented in freely available computer algebra system SINGULAR. It relies on Gröbner bases in noncommutative PBW algebras. The algorithm allows us to exhibit some explicit non-trivial examples and to support some existing conjectures.

math.AG↗

arXiv2025

Modular Algorithms For Computing Gröbner Bases in Free Algebras

In this work, we extend modular techniques for computing Gröbner bases involving rational coefficients to (two-sided) ideals in free algebras. We show that the infinite nature of Gröbner bases in this setting renders the classical approach infeasible. Therefore, we propose a new method that relies on signature-based algorithms. Using the data of signatures, we can overcome the limitations of the classical approach and obtain a practical modular algorithm. Moreover, the final verification test in this setting is both more general and more efficient than the classical one. We provide a first implementation of our modular algorithm in SageMath. Initial experiments show that the new algorithm can yield significant speedups over the non-modular approach.

cs.SC↗

arXiv2021

Computing Free Non-commutative Groebner Bases over Z with Singular:Letterplace

With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z . We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras.The extension of Groebner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance.Groebner and Groebner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Groebner bases between free algebras and algebras, close to commutative.Even the process of the formation of critical pairs has to be reengineered, together with the implementing the criteria for their quick discarding.We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Groebner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.

math.RA↗

arXiv2021

Gröbner bases for fusion products

We provide a new approach towards the analysis of the fusion products defined by B.~Feigin and S.~Loktev in the representation theory of (truncated) current Lie algebras. We understand the fusion product as a degeneration using Gröbner theory of non-commutative algebras and outline a strategy on how to prove a conjecture about the defining relations for the fusion product of two evaluation modules. We conclude with following this strategy for $\mathfrak{sl}_2(\mathbb{C}[t]) $ and hence provide yet another proof for the conjecture in this case.

math.RT↗

arXiv2020

Constructive Arithmetics in Ore Localizations Enjoying Enough Commutativity

This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity.

math.RA↗

arXiv2019

Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

math.QA↗

arXiv2019

Left saturation closure for Ore localizations

In this paper, we introduce the notion of LSat, the left saturation closure of a subset of a module at a subset of the base ring, which generalizes multiple important concepts related to Ore localization. We show its significance in finding a saturated normal form for left Ore sets as well as in characterizing the units of a localized ring. Furthermore, LSat encompasses the notion of local closure of submodules and ideals from the realm of algebraic analysis, where it describes the result of extending a submodule or ideal from a ring to its localization and contracting it back again.

math.RA↗

arXiv2017

Constructive Arithmetics in Ore Localizations of Domains

For a non-commutative domain $R$ and a multiplicatively closed set $S$ the (left) Ore localization of $R$ at $S$ exists if and only if $S$ satisfies the (left) Ore property. Since the concept has been introduced by Ore back in the 1930's, Ore localizations have been widely used in theory and in applications. We investigate the arithmetics of the localized ring $S^{-1}R$ from both theoretical and practical points of view. We show that the key component of the arithmetics is the computation of the intersection of a left ideal with a submonoid $S$ of $R$. It is not known yet, whether there exists an algorithmic solution of this problem in general. Still, we provide such solutions for cases where $S$ is equipped with additional structure by distilling three most frequently occurring types of Ore sets. We introduce the notion of the (left) saturation closure and prove that it is a canonical form for (left) Ore sets in $R$. We provide an implementation of arithmetics over the ubiquitous $G$-algebras in \textsc{Singular:Plural} and discuss questions arising in this context. Numerous examples illustrate the effectiveness of the proposed approach.

math.RA↗

arXiv2017

Modular Techniques For Noncommutative Gröbner Bases

In this note, we extend modular techniques for computing Gröbner bases from the commutative setting to the vast class of noncommutative $G$-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gröbner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system {\sc{Singular}} and give timings to compare its performance with that of other instances of Buchberger's algorithm, testing examples from $D$-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.

math.RA↗

arXiv2016

Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-graded rings. We show, that factorization of homogeneous polynomials can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail. Furthermore, we will show for homogeneous polynomials that irreducibility in the polynomial first Weyl algebra also implies irreducibility in the rational one, which is of interest for practical reasons. We report on our implementation in the computer algebra system \textsc{Singular}. It outperforms for homogeneous polynomials currently available implementations dealing with factorization in the first Weyl algebra both in speed and elegancy of the results.

cs.SC↗

arXiv2016

A Factorization Algorithm for G-Algebras and Applications

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is any $G$-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gröbner basis algorithm for $G$-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from $\mathcal{G}$. Additionally, it is possible to include inequality constraints for ideals in the input.

math.RA↗

arXiv2015

On Noncommutative Finite Factorization Domains

A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra. We show that if $A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite then $A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration.

math.RA↗

arXiv2014

Factoring Differential Operators in n Variables

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is ZZ^n-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in commodity computer algebra systems on nontrivial examples.

cs.SC↗

arXiv2013

Quantum Drinfeld Hecke Algebras

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincare-Birkhoff-Witt property using the theory of noncommutative Groebner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincare-Birkhoff-Witt conditions.

math.RA↗

arXiv2013

SymbolicData:SDEval - Benchmarking for Everyone

In this paper we will present SDeval, a software project that contains tools for creating and running benchmarks with a focus on problems in computer algebra. It is built on top of the Symbolic Data project, able to translate problems in the database into executable code for various computer algebra systems. The included tools are designed to be very flexible to use and to extend, such that they can be utilized even in contexts of other communities. With the presentation of SDEval, we will also address particularities of benchmarking in the field of computer algebra. Furthermore, with SDEval, we provide a feasible and automatizable way of reproducing benchmarks published in current research works, which appears to be a difficult task in general due to the customizability of the available programs. We will simultaneously present the current developments in the Symbolic Data project.

cs.SC↗

arXiv2012

Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra

In this paper we introduce an algebra embedding $ι:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * Σ$ defined by the commutative polynomial ring $P = K[X\times N^*]$ and by the monoid $Σ= < σ>$ generated by a suitable endomorphism $σ:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P σ^i$ and $σ:P \to P$ an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of $P$. Moreover, using a suitable grading for the algebra $P$ compatible with the action of $Σ$, we obtain a bijective correspondence, preserving Gröbner bases, between graded $Σ$-invariant ideals of $P$ and a class of graded two-sided ideals of $S$. By means of the embedding $ι$ this results in the unification, in the graded case, of the Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$ fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring $S$ and in a finite number of steps up to some fixed degree.

math.RA↗

arXiv2011

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{ö}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.

math.RA↗

arXiv2011

Certifying solutions to square systems of polynomial-exponential equations

Smale's alpha-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented in alphaCertified. Examples are presented to demonstrate this certification algorithm.

math.NA↗