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157
A New Criterion for Normal Form Algorithms
 Proc. AAECC, volume 1719 of LNCS
, 1999
"... In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This cr ..."
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Cited by 65 (19 self)
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In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of Spolynomials. It leads to a new algorithm for constructing the multiplicative structure of a zerodimensional algebra. Described in terms of intrinsic operations on vector spaces in the ring of polynomials, this algorithm extends naturally to Laurent polynomials.
Involutive Bases of Polynomial Ideals
, 1999
"... 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 ..."
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Cited by 54 (13 self)
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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 selfconsistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and nonmultiplicative. 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 Gröbner 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.
Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms
, 1996
"... We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a consta ..."
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Cited by 53 (16 self)
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We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...
A Completion Procedure for Computing a Canonical Basis for a kSubalgebra
 IN COMPUTERS AND MATHEMATICS
, 1989
"... A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewritin ..."
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Cited by 37 (0 self)
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A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting concepts. A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a ksubalgebra to 0 and generating unique normal forms for the equivalence classes generated by a ksubalgebra. In contrast to Shannon and Sweedler's approach using tag variables, this approach is direct. One of the limitations of the approach however is that the procedure may not terminate for some term orderings thus giving an infinite canonical basis. The procedure is illustrated using examples.
"One sugar cube, please" or Selection strategies in the Buchberger algorithm
 Proceedings of the ISSAC'91, ACM Press
, 1991
"... In this paper we describe some experimental findings on selection strategies for Gröbner basis computation with the Buchberger algorithm. In particular, the results suggest that the "sugar flavor" of the "normal selection", implemented first in CoCoA, then in AlPI, and now in SCR ..."
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Cited by 37 (2 self)
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In this paper we describe some experimental findings on selection strategies for Gröbner basis computation with the Buchberger algorithm. In particular, the results suggest that the "sugar flavor" of the "normal selection", implemented first in CoCoA, then in AlPI, and now in SCRATCHPADII, is the best choice for a selection strategy. It has to be combined with the "straightforward" simplification strategy and with a special form of the GebauerMöller criteria to obtain the best results. The idea of the "sugar flavor" is the following: the Buchberger algorithm for homogeneous ideals, with degreecompatible term ordering and normal selection strategy, usually works fine. Homogenizing the basis of the ideal is good for the strategy, but bad for the basis to be computed. The sugar flavor computes, for every polynomial in the course of the algorithm, "the degree that it would have if computed with the homogeneous algorithm", and uses this phantom degree (the sugar) only for the selection strategy. We have tested several examples with different selection strategies, and the sugar flavor has proved to be always the best choice or very near to it. The comparison between the different variants of the sugar flavor has been made, but the results are up to now inconclusive. We include a complete deterministic description of the Buchberger algorithm as it was used in our experiments.
Gröbner bases for operads
 DUKE MATH. J
, 2009
"... We define a new monoidal category on collections (shuffle ..."
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Cited by 31 (6 self)
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We define a new monoidal category on collections (shuffle
Differential EliminationCompletion Algorithms for DAE and PDAE
, 1995
"... DAE and PDAE are systems of ordinary and partial differentialalgebraic equations with constraints. They occur frequently in applications such as constrained multibody mechanics, spacecraft control and incompressible fluid dynamics. A dae has differential index r if a minimum of r +1 differentiatio ..."
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Cited by 26 (12 self)
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DAE and PDAE are systems of ordinary and partial differentialalgebraic equations with constraints. They occur frequently in applications such as constrained multibody mechanics, spacecraft control and incompressible fluid dynamics. A dae has differential index r if a minimum of r +1 differentiations of it are required before no new constraints are obtained. While dae of low differential index (0 or 1) are generally easier to solve numerically, higher index dae present severe difficulties. Reich, Rabier and Rheinboldt have presented a geometric theory and an algorithm for reducing dae of high differential index to dae of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for dae of low differential index. We show that for analytic autonomous first order dae this algorithm is equivalent to the CartanKuranishi algorithm for completing a system of differential equations to involutive form. The CartanKuranishi algorithm has the advantage tha...
Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations
, 1996
"... this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equatio ..."
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Cited by 25 (0 self)
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this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.