arXiv subjects

Vladimir P. Gerdt

Publications and source records attributed to Vladimir P. Gerdt.

32 records · Page 2Linked to original sources

arXiv2005

An Algorithm for Constructing Polynomial Systems Whose Solution Space Characterizes Quantum Circuits

An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z_2 for the polynomial system associated with the circuit. To determine the number of solutions in Z_2 for the output polynomial system, one can use the Groebner bases method and the relevant algorithms for computing Groebner bases.

quant-ph↗

arXiv2005

A Maple Package for Computing Groebner Bases for Linear Recurrence Relations

A Maple package for computing Groebner bases of linear difference ideals is described. The underlying algorithm is based on Janet and Janet-like monomial divisions associated with finite difference operators. The package can be used, for example, for automatic generation of difference schemes for linear partial differential equations and for reduction of multiloop Feynman integrals. These two possible applications are illustrated by simple examples of the Laplace equation and a one-loop scalar integral of propagator type

cs.SC↗

arXiv2005

On Computation of Groebner Bases for Linear Difference Systems

In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The algorithm is an adaptation to difference ideals of our polynomial algorithm based on Janet-like reductions.

math-ph↗

arXiv2005

Janet Bases of Toric Ideals

In this paper we present a version of the general polynomial involutive algorithm for computing Janet bases specialized to toric ideals. The relevant data structures are Janet trees which provide a very fast search for a Janet divisor. We broach also efficiency issues in view of application of the algorithm presented to computation of toric ideals.

math.AC↗

arXiv2005

Involutive Algorithms for Computing Groebner Bases

In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Groebner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be output without any extra computational costs. We also discuss some accounts of experimental superiority of the involutive algorithm over Buchberger's algorithm.

math.AC↗

arXiv2005

Groebner Bases in Perturbative Calculations

In this paper we outline the most general and universal algorithmic approach to reduction of loop integrals to basic integrals. The approach is based on computation of Groebner bases for recurrence relations derived from the integration by parts method. In doing so we consider generic recurrence relations when propagators have arbitrary integer powers treated as symbolic variables (indices) for the relations.

hep-ph↗

arXiv2003

New exact solutions for polynomial oscillators in large dimensions

A new type of exact solvability is reported. We study the general central polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's partial exact solvability conditions (this means that they possess a harmonic-oscillator-like wave function proportional to a polynomial of any integer degree N). Working in the space of a very large dimension D for simplicity, we reveal that in contrast to the usual version of the model in finite dimensions (requiring a purely numerical treatment of the Magyari's constraints), our large D problem acquires an explicit, closed form solution at all N and up to q = 5 at least. This means that our effective secular polynomials (generated via the standard technique of Groebner bases) happen to be all fully factorizable in an utterly mysterious manner (mostly, over integers).

math-ph↗

arXiv2000

On the Relation Between Pommaret and Janet Bases

In this paper the relation between Pommaret and Janet bases of polynomial ideals is studied. It is proved that if an ideal has a finite Pommaret basis then the latter is a minimal Janet basis. An improved version of the related algorithm for computation of Janet bases, initially designed by Zharkov, is described. For an ideal with a finite Pommaret basis, the algorithm computes this basis. Otherwise, the algorithm computes a Janet basis which need not be minimal. The obtained results are generalized to linear differential ideals.

math.AC↗

arXiv2000

Completion of Linear Differential Systems to Involution

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutive algorithms. We present an algorithm for computation of a minimal involutive differential basis. Its correctness and termination hold for any constructive and noetherian involutive division. As two important applications we consider posing of an initial value problem for a linear differential system providing uniqueness of its solution and the Lie symmetry analysis of nonlinear differential equations. In particular, this allows to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.

math.AP↗

arXiv2000

Involutive Division Technique: Some Generalizations and Optimizations

In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Groebner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on properties of particular involutive divisions two computational optimizations are suggested. One of them consists in a special choice of the completion ordering. Another optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm.

math.AC↗

arXiv1999

Constrained Hamiltonian Systems and Groebner Bases

In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods of commutative algebra based on the use of Groebner bases. As it is shown, this makes the classical Dirac method fully algorithmic. The underlying algorithm implemented in Maple is presented and some illustrative examples are given.

math.NA↗

arXiv1999

Minimal Involutive Bases

In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Groebner basis.

math.AC↗

arXiv1999

Involutive Bases of Polynomial Ideals

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Groebner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchberger's chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.

math.AC↗

arXiv1999

Construction of Involutive Monomial Sets for Different Involutive Divisions

We consider computational and implementation issues for the completion of monomial sets to involution using different involutive divisions. Every of these divisions produces its own completion procedure. For the polynomial case it yields an involutive basis which is a special form of a Groebner basis, generally redundant. We also compare our Mathematica implementation of Janet division to an implementation in C.

math.AC↗