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33
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 40 (11 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.
Algebraic and Geometric Reasoning using Dixon Resultants
 IN ACM ISSAC 94
, 1994
"... Dixon's method for computing multivariate resultants by simultaneously eliminating many variables is reviewed. The method is found to be quite restrictive because often the Dixon matrix is singular, and the Dixon resultant vanishes identically yielding no information about solutions for many algebra ..."
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Cited by 38 (16 self)
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Dixon's method for computing multivariate resultants by simultaneously eliminating many variables is reviewed. The method is found to be quite restrictive because often the Dixon matrix is singular, and the Dixon resultant vanishes identically yielding no information about solutions for many algebraic and geometry problems. We extend Dixon's method for the case when the Dixon matrix is singular, but satisfies a condition. An efficient algorithm is developed based on the proposed extension for extracting conditions for the existence of affine solutions of a finite set of polynomials. Using this algorithm, numerous geometric and algebraic identities are derived for examples which appear intractable with other techniques of triangulation such as the successive resultant method, the Grobner basis method, Macaulay resultants and Characteristic set method. Experimental results suggest that the resultant of a set of polynomials which are symmetric in the variables is relatively easier to comp...
Efficient Algorithms and Bounds for WuRitt Characteristic Sets
 in Proc. of MEGA 90
, 1990
"... The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu WenTsun, in the late seventies. Since then WuRitt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated T ..."
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Cited by 31 (5 self)
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The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu WenTsun, in the late seventies. Since then WuRitt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated Theorem Proving in Elementary Geometries and Computer Vision. In this paper, we present optimal algorithms for computing a characteristic set with simpleexponential sequential and polynomial parallel time complexities. These algorithms are derived, via linear algebra, from simpleexponential degree bounds for a characteristic set. The degree bounds are obtained by using the recent effective version of Hilbert's Nullstellensatz, due to D. Brownawell and J. Koll'ar, and a version of Bezout's Inequality, due to J. Heintz. 1. Introduction In the late forties, J.F. Ritt, in his now classic book Differential Algebra [28], introduced an effective process to construct a triangular set of equations f...
Symbolic Computation: Computer Algebra and Logic
 Frontiers of Combining Systems, Applied Logic Series
, 1996
"... In this paper we present our personal view of what should be the next step in the development of symbolic computation systems. The main point is that future systems should integrate the power of algebra and logic. We identify four gaps between the future ideal and the systems available at present: t ..."
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Cited by 26 (3 self)
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In this paper we present our personal view of what should be the next step in the development of symbolic computation systems. The main point is that future systems should integrate the power of algebra and logic. We identify four gaps between the future ideal and the systems available at present: the logic, the syntax, the mathematics, and the prover gap, respectively. We discuss higher order logic without extensionality and with set theory as a subtheory as a logic frame for future systems and we propose to start from existing computer algebra systems and proceed by adding new facilities for closing the syntax, mathematics, and the prover gaps. Mathematica seems to be a particularly suitable candidate for such an approach. As the main technique for structuring mathematical knowledge, mathematical methods (including algorithms), and also mathematical proofs, we underline the practical importance of functors and show how they can be naturally embedded into Mathematica. 1 The Next Goal ...
Sweeping of Threedimensional Objects
 ComputerAided Design
, 1989
"... this paper addresses is the following: given an arbitrary threedimensional object, which moves along an arbitrary path (possibly rotating as it does so), to compute the volume swept out by the solid object as it moves, or in other words, to find the new solid volume which represents all of the poin ..."
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Cited by 23 (0 self)
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this paper addresses is the following: given an arbitrary threedimensional object, which moves along an arbitrary path (possibly rotating as it does so), to compute the volume swept out by the solid object as it moves, or in other words, to find the new solid volume which represents all of the points in space which the object has occupied at some time during the motion
Solving geometric constraint systems. II. a symbolic approach and decision of rcconstructibility
 ComputerAided Design
, 1998
"... Abstract. This paper reports a geometric constraint solving approach based on symbolic computation. With this approach, we can compute robust numerical solutions for a set of equations. We give complete methods of deciding whether the constraints are independent and whether a constraint system is ov ..."
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Cited by 23 (3 self)
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Abstract. This paper reports a geometric constraint solving approach based on symbolic computation. With this approach, we can compute robust numerical solutions for a set of equations. We give complete methods of deciding whether the constraints are independent and whether a constraint system is overconstraint. Also, overconstrainted systems can be handled naturally. Based on symbolic computation, we also give a decision procedure for the problem of deciding whether a constrainted diagram can be constructed with ruler and compass (rcconstructibility). 1.
Comparison of Various Multivariate Resultant Formulations
 PROC. INTERNAT. SYMP. SYMBOLIC ALGEBRAIC COMPUT. ISSAC '95
, 1995
"... Three most important resultant formulations are the Macaulay, Dixon and sparse resultant formulations. For most polynomial systems, however, the matrices constructed in these formulations become singular and the projection operator vanishes identically. In such cases, perturbation techniques for Mac ..."
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Cited by 22 (9 self)
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Three most important resultant formulations are the Macaulay, Dixon and sparse resultant formulations. For most polynomial systems, however, the matrices constructed in these formulations become singular and the projection operator vanishes identically. In such cases, perturbation techniques for Macaulay formulation such as generalized characteristic polynomial (GCP) and a method based on rank submatrix computation (RSC), applicable to all three formulations, can be used, giving four methods, Macaulay/GCP, Macaulay/RSC, Dixon/RSC and Sparse/RSC, for computing nontrivial projection operators. In this paper, these four methods are compared. It is shown that the Dixon matrix is (by a factor up to O(e n ) for a certain class) smaller than the sparse resultant matrix which is (by a factor up to O(e n ) for a certain class) smaller than the Macaulay matrix. Empirical results confirm that Dixon/RSC is the most efficient, followed by Sparse/RSC then Macaulay/RSC and finally Macaulay/GCP, ...
Hierarchical Arc Consistency for Disjoint Real Intervals in Constraint Logic Programming
 COMPUTATIONAL INTELLIGENCE
, 1992
"... There have been many proposals for adding sound implementations of numeric processing to Prolog. This paper describes an approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language. Echidna uses consistency algorithms which can a ..."
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Cited by 19 (0 self)
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There have been many proposals for adding sound implementations of numeric processing to Prolog. This paper describes an approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language. Echidna uses consistency algorithms which can actively process a wider variety of numeric constraints than most other CLP systems, including constraints containing some common nonlinear functions. A unique feature of Echidna is that it implements domains for realvalued variables with hierarchical data structures and exploits this structure using a hierarchical arc consistency algorithm specialized for numeric constraints. This gives Echidna two advantages over other systems. First, the union of disjoint intervals can be represented directly. Other approaches require trying each disjoint interval in turn during backtrack search. Second, the hierarchical structure facilitates varying the precision of constraint processing. Consequently...
Minimal involutive bases
 Math. Comp. Simul
, 1998
"... In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Gröbner 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 ..."
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Cited by 15 (5 self)
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In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Gröbner 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 nonmultiplicative. This partition gives thereby the selfconsistent computational procedure for constructing an involutive basis by performing nonmultiplicative 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 Gröbner basis. 1
Involutive Division Technique: Some Generalizations and Optimizations
, 1998
"... 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 Gröbner basis of a polynom ..."
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Cited by 14 (8 self)
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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 Gröbner 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. 1