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415
ALGEBRAIC GEOMETRY
"... Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is ..."
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Cited by 521 (6 self)
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Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is
A New Efficient Algorithm for Computing Gröbner Bases (F4)
 IN: ISSAC ’02: PROCEEDINGS OF THE 2002 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 2002
"... This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduc ..."
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Cited by 366 (57 self)
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This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations.
On the Combinatorial and Algebraic Complexity of Quantifier Elimination
, 1996
"... In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of th ..."
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Cited by 231 (29 self)
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In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated. Another new feature is that the degrees of the polynomials in the equivalent quantifierfree formula that is output, are independent of the number of input polynomials. As special cases of this algorithm, new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 103 (12 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
PicardFuchs Equations and Mirror Maps For Hypersurfaces
 IN ESSAYS ON MIRROR MANIFOLDS, ED. S.T.YAU, INTERNATIONAL PRESS
, 1992
"... We describe a strategy for computing Yukawa couplings and the mirror map, based on the PicardFuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes [5] in the case of quintic hypersurfaces.) We then explain a technique of Griffiths [14] which can ..."
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Cited by 102 (5 self)
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We describe a strategy for computing Yukawa couplings and the mirror map, based on the PicardFuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes [5] in the case of quintic hypersurfaces.) We then explain a technique of Griffiths [14] which can be used to compute the PicardFuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al. [5]). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weighted projective spaces. Some of these predictions have been confirmed by classical techniques in algebraic geometry.
An introduction to commutative and noncommutative Gröbner bases
 Theoretical Computer Science
, 1994
"... In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational ..."
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Cited by 97 (3 self)
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In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational
A Geometric Constraint Solver
, 1995
"... We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction directed algebraic approach, and achieves interact ..."
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Cited by 72 (10 self)
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We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction directed algebraic approach, and achieves interactive speed. We describe the architecture of the solver and its basic capabilities. Then, we discuss in detail how to extend the scope of the solver, with special emphasis placed on the theoretical and human factors involved in finding a solution  in an exponentially large search space  so that the solution is appropriate to the application and the way of finding it is intuitive to an untrained user. 1 Introduction Solving a system of geometric constraints is a problem that has been considered by several communities, and using different approaches. For example, the symbolic computation community has considered the general problem, in the Supported in part by ONR contract N0001490J...
Polynomial interpolation in several variables
, 2000
"... This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique inter ..."
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Cited by 71 (6 self)
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This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.
A Geometric Buchberger Algorithm for Integer Programming
 Mathematics of Operations Research
, 1995
"... Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution α to a program in this family, there i ..."
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Cited by 63 (11 self)
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Let IP denote the family of integer programs of the form Min cx : Ax = b, x &isin; N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution &alpha; to a program in this family, there is at least one element g in this set such that &alpha;  g has an improved cost value as compared to &alpha;. We describe a unique minimal test set for this family called the reduced Gröbner basis of IP. An algorithm for its construction is presented which we call a Geometric Buchberger Algorithm for integer programming. We show how an integer program may be solved using this test set and examine some geometric properties of elements in the set. The reduced Grobner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A &isin; Z^(n2)&times;n of full row rank.