Viktor Levandovskyy | University of Kassel (original) (raw)

Papers by Viktor Levandovskyy

arXiv (Cornell University), Mar 12, 2020

Texts & Monographs in Symbolic Computation, 2012

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-... more With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z〈X〉. 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 Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-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 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 rev...

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix usin... more 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 pa...

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl alg... more We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and q-Weyl algebra, which are both viewed as a 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 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.

A domain R is said to have the finite factorization property if every nonzero non-unit element of... more 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.

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, ... more 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 ∈G, where 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 G. Additionally, it is possible to include inequality constraints for ideals in the input.

Linear exact modeling is a problem coming from system identification: Given a set of observed tra... more Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.

Journal of Symbolic Computation, 2009

We overview numerous algorithms in computational D-module theory together with the theoretical ba... more We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

This paper continues a research program on constructive investigations of non-commutative Ore loc... more 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 ...

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applica-tions. I... more Bernstein-Sato polynomial of a hypersurface is an important object with numerous applica-tions. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called checkRoot for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and b-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A’Campo and spectral numbers. We also present approaches to the logarith-mic comparison problem and the intersection homology D-module. Several applications are presen...

In this paper we introduce an algebra embedding ι:K< X >→ S from the free associative algeb... more In this paper we introduce an algebra embedding ι:K< X >→ 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× N^*] and by the monoid Σ = < σ > generated by a suitable endomorphism σ:P→ 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 = ⊕_i S_i with S_i = P σ^i and σ:P → 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 ...

We consider polynomially nonlinear state-space systems and given algebraic varieties. A variety V... more We consider polynomially nonlinear state-space systems and given algebraic varieties. A variety V is said to be controlled invariant w.r.t. a given system if we can find a polynomial state feedback law that causes the closed loop system to have V as an invariant set. If this task can be achieved by a polynomial output feedback law, V is called controlled and conditioned invariant. This concept leads to the problem of determining the intersection of a certain (affine) submodule of a free module over a polynomial ring with a free module over a subalgebra of this ring. We suggest various approaches to do this and to decide whether a variety is controlled and conditioned invariant, and if so, to compute all output feedback laws achieving the task.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semid... more 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.

In this paper we discuss three symbolic approaches for the generation of a finite difference sche... more In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent and discuss the applicability of them to nonlinear PDE's as well as to the case of variable coefficients. Moreover, we systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive the conditions on the von Neumann stability of a difference s cheme for a linear PDE with constant coefficients. For stable schemes we demonst rate algorithmic and symbolic approach to handle both continuous and discrete di spersion. We present an implementation of tools for generation of schemes, which rely on Gr\"obner basis, in the system SINGULAR and present numerous e xamples, computed with our implementation. In the stability analysis, we use the system MATHEMATICA for cylindrical algebraic decomposition.

In this paper we present a very general way to generate finite difference schemes of arbitrary pa... more In this paper we present a very general way to generate finite difference schemes of arbitrary partial differential equations analytically. This approach uses the concept of polynomial rings and Gröbner bases. A criterion for the existence of a scheme for a partial differential equation with some arbitrary approximation rules is given.

ArXiv, 2010

We overview numerous algorithms in computational DDD-module theory together with the theoretical ... more We overview numerous algorithms in computational DDD-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, 2018

We continue the investigations of the constructivity of arithmetics within non-commutative Ore lo... more We continue the investigations of the constructivity of arithmetics within non-commutative Ore localizations, initiated in our 2017 ISSAC paper, where 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. We provide algorithms to compute such closures for certain non-commutative rings with respect to Ore sets with enough commutativity.

arXiv (Cornell University), Mar 12, 2020

Texts & Monographs in Symbolic Computation, 2012

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-... more With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z〈X〉. 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 Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-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 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 rev...

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix usin... more 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 pa...

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl alg... more We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and q-Weyl algebra, which are both viewed as a 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 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.

A domain R is said to have the finite factorization property if every nonzero non-unit element of... more 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.

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, ... more 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 ∈G, where 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 G. Additionally, it is possible to include inequality constraints for ideals in the input.

Linear exact modeling is a problem coming from system identification: Given a set of observed tra... more Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.

Journal of Symbolic Computation, 2009

We overview numerous algorithms in computational D-module theory together with the theoretical ba... more We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

This paper continues a research program on constructive investigations of non-commutative Ore loc... more 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 ...

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applica-tions. I... more Bernstein-Sato polynomial of a hypersurface is an important object with numerous applica-tions. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called checkRoot for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and b-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A’Campo and spectral numbers. We also present approaches to the logarith-mic comparison problem and the intersection homology D-module. Several applications are presen...

In this paper we introduce an algebra embedding ι:K< X >→ S from the free associative algeb... more In this paper we introduce an algebra embedding ι:K< X >→ 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× N^*] and by the monoid Σ = < σ > generated by a suitable endomorphism σ:P→ 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 = ⊕_i S_i with S_i = P σ^i and σ:P → 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 ...

We consider polynomially nonlinear state-space systems and given algebraic varieties. A variety V... more We consider polynomially nonlinear state-space systems and given algebraic varieties. A variety V is said to be controlled invariant w.r.t. a given system if we can find a polynomial state feedback law that causes the closed loop system to have V as an invariant set. If this task can be achieved by a polynomial output feedback law, V is called controlled and conditioned invariant. This concept leads to the problem of determining the intersection of a certain (affine) submodule of a free module over a polynomial ring with a free module over a subalgebra of this ring. We suggest various approaches to do this and to decide whether a variety is controlled and conditioned invariant, and if so, to compute all output feedback laws achieving the task.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semid... more 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.

In this paper we discuss three symbolic approaches for the generation of a finite difference sche... more In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent and discuss the applicability of them to nonlinear PDE's as well as to the case of variable coefficients. Moreover, we systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive the conditions on the von Neumann stability of a difference s cheme for a linear PDE with constant coefficients. For stable schemes we demonst rate algorithmic and symbolic approach to handle both continuous and discrete di spersion. We present an implementation of tools for generation of schemes, which rely on Gr\"obner basis, in the system SINGULAR and present numerous e xamples, computed with our implementation. In the stability analysis, we use the system MATHEMATICA for cylindrical algebraic decomposition.

In this paper we present a very general way to generate finite difference schemes of arbitrary pa... more In this paper we present a very general way to generate finite difference schemes of arbitrary partial differential equations analytically. This approach uses the concept of polynomial rings and Gröbner bases. A criterion for the existence of a scheme for a partial differential equation with some arbitrary approximation rules is given.

ArXiv, 2010

We overview numerous algorithms in computational DDD-module theory together with the theoretical ... more We overview numerous algorithms in computational DDD-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, 2018

We continue the investigations of the constructivity of arithmetics within non-commutative Ore lo... more We continue the investigations of the constructivity of arithmetics within non-commutative Ore localizations, initiated in our 2017 ISSAC paper, where 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. We provide algorithms to compute such closures for certain non-commutative rings with respect to Ore sets with enough commutativity.