Involutive bases of polynomial ideals

  title={Involutive bases of polynomial ideals},
  author={Vladimir P. Gerdt and Yuri A. Blinkov},
  journal={Mathematics and Computers in Simulation},

Minimal involutive bases

An algorithm for construction of minimal involutive polynomial bases which are Grobner bases of the special form is presented which provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering.

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

  • J. Apel
  • Mathematics
    J. Symb. Comput.
  • 1998
This theory introduces the lattice of so-called involutive divisions and defines the admissibility of such an involutive division for a given set of terms and presents a new approach for building a general theory of involutive bases of polynomial ideals.

On Connection Between Constructive Involutive Divisions and Monomial Orderings

It is proven that Janet division has the advantage in the minimal involutive basis size of the class of ≻ -divisions which possess many good properties of Janet division and can be considered as its analogs for orderings different from the lexicographic one.

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

Involutive Algorithms for Computing

An efficient involutive algorithm based on the concept of involutive monomial division which restricts the conventional division in a certain way and which can be output without any extra computational costs is described.

Term-ordering free involutive bases

Involutive Algorithms for Computing Groebner Bases

An involutive algorithm for construct- ing Gröbner bases of polynomial ideals based on the concept of involutive monomial division which restricts the conventional division in a certain way is described.

Involutive Division Techniques: Some Generalizations and Optimizations

A new class of involutive divisions induced by certain orderings of monomials is considered. It is proved that these divisions are Noetherian and constructive. Therefore, each of them allows one to

Completion of Linear Differential Systems to Involution

This paper considers posing of an initial value problem for a linear differential system providing uniqueness of its solution and Lie symmetry analysis of nonlinear differential equations to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.



Solving zero-dimensional involutive systems

It turns out that when the involutive basis exists (without change of variables), it can be computed considerably faster by this algorithm than the minimal standard basis by Buchberger’s algorithm.

A Gröbner Approach to Involutive Bases

  • J. Apel
  • Mathematics
    J. Symb. Comput.
  • 1995
A pure algebraic foundation of involutive bases of Pommaret type is presented, which turns out to be generalized left Grobner bases of ideals in the commutative polynomial ring with respect to a non-commutative grading.

Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory

Problems connected with ideals generated by finite sets F of multivariate polynomials occur, as mathematical subproblems, in various branches of systems theory, see, for example, [5]. The method of

Groebner Bases and Differential Algebra

The notion of A-Gr6bner basis arises in natural way but unfortunately there are finitely A-generated ~-ideels that have no such a basis, so this approach cannot be used to solve the membership problem.

Non-Commutative Gröbner Bases in Algebras of Solvable Type

Gröbner bases and involutive methods for algebraic and differential equations

  • V. Gerdt
  • Mathematics, Computer Science
  • 1997

A criterion for detecting unnecessary reductions in the construction of Groebner bases

A new criterion is presented that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals and allows to derive a realistic upper bound for the degrees of the polynomials in the GroBner-Bases computed by the algorithm in the case of poylemials in two variables.

Standard Bases of Differential Ideals

A new definition of standard bases of differential ideals, allowing more general orderings than the previous one, given by Giuseppa Carra-Ferro, is introduced, valid for algebraic ideals, canonical bases of subalgebras, etc.

Gröbner bases - a computational approach to commutative algebra

This chapter discusses linear algebra in Residue Class Rings in Vector Spaces and Modules, and first applications of Gr bner Bases.