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Polar varieties and efficient real elimination
 MATH. Z
, 2001
"... Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations th ..."
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Cited by 28 (12 self)
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Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.
Generalized polar varieties: Geometry and algorithms
, 2004
"... Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motiva ..."
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Cited by 25 (7 self)
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Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in [2]. As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar varietiy, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in
On the TimeSpace Complexity of Geometric Elimination Procedures
, 1999
"... In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new ge ..."
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Cited by 23 (16 self)
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In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input.
Kronecker's Smart, Little Black Boxes
"... This paper is devoted to the complexity analysis of certain uniformity properties owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which is parsimonious with respect to branchings and division ..."
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Cited by 17 (5 self)
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This paper is devoted to the complexity analysis of certain uniformity properties owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which is parsimonious with respect to branchings and divisions must necessarily have a nonpolynomial sequential time complexity, even if highly ecient data structures (as e.g. the arithmetic circuit encoding of polynomials) are used.
The Hardness of Polynomial Equation Solving
, 2003
"... Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic seq ..."
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Cited by 17 (9 self)
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Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects.
A concise proof of the Kronecker polynomial system solver from scratch
, 2006
"... Abstract. Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms are based on Gröbner bases, resultants, Macaulay matrices, or triangular decompositions. In all these algorithms, multivariat ..."
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Cited by 9 (1 self)
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Abstract. Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms are based on Gröbner bases, resultants, Macaulay matrices, or triangular decompositions. In all these algorithms, multivariate polynomials are expanded in a monomial basis, and the computations mainly reduce to linear algebra. The major drawback of these techniques is the exponential explosion of the size of the polynomials needed to represent highly positive dimensional solution sets. Alternatively, the “Kronecker solver ” uses data structures to represent the input polynomials as the functions that compute their values at any given point. In this paper we present the first selfcontained and student friendly version of the Kronecker solver, with a substantially simplified proof of correctness. In addition, we enhance the solver in order to compute the multiplicities of the zeros without any extra cost.
Functional Programming Concepts and Straightline Programs in Computer Algebra
 Mathematics and Computers in Simulation
, 2003
"... In this paper we present MILONGA, a language based on functional programming concepts, which was designed for the implementation of a new generation of non{ term{rewriting elimination algorithms for multivariate polynomial solving (Giusti et al. [J. Pure Appl. Alg. 124, pp. 146, 1998], Giusti et al ..."
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Cited by 3 (2 self)
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In this paper we present MILONGA, a language based on functional programming concepts, which was designed for the implementation of a new generation of non{ term{rewriting elimination algorithms for multivariate polynomial solving (Giusti et al. [J. Pure Appl. Alg. 124, pp. 146, 1998], Giusti et al. [J. Pure Appl. Alg. 117/118, pp. 277317, 1997]; see also Heintz et al. [To appear in Appl. Alg. in Eng., Commun. & Comput., 2000], Giusti et al. [To appear in J. Complexity, 2000]).
The differential Hilbert function of a differential rational mapping can be computed in polynomial time
 In Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation
, 2002
"... We present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in order to locally invert the differential mappi ..."
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Cited by 1 (0 self)
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We present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in order to locally invert the differential mapping under consideration. The arithmetic complexity of this algorithm is polynomial in the input size.
Equations for Polar Varieties and Efficient Real Elimination
, 2000
"... Let V 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 ; : : : ; f p . This paper is devoted to the algorithmic problem of finding efficiently for each connected component of V 0 a representative point. For this purpose we exhibit explicit polynomial equa ..."
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Let V 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 ; : : : ; f p . This paper is devoted to the algorithmic problem of finding efficiently for each connected component of V 0 a representative point. For this purpose we exhibit explicit polynomial equations which describe for generic variables the polar varieties of V 0 of all dimensions. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 ; : : : ; f p and in a suitably introduced geometric (extrinsic) parameter, called the degree of the real interpretation of the given equation system f 1 ; : : : ; f p . Keywords: Real polynomial equation solving, polar variety, geometric degree, arithmetic circuit, arithmetic network, complexity 1 Research partially supported by the following German, French, Spanish and Argentinian grants: BA 1257/41 (DFG), ARG 018/98 INF (BMBF), ECOS A99E06, DGI...