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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 26 (14 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 12 (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.
1Departamento de Ciencias de la Computación
, 2003
"... Dedicated to Michel Demazure... Il est fréquent, devant un problème concret, de trouver un théorème qui “s’applique presque”.... Le rôle des contreexemples est justement de délimiter le possible, et ce n’est pas par perversité (ou en tout cas pas totalement) que les textes mathématiques exhibent de ..."
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Dedicated to Michel Demazure... Il est fréquent, devant un problème concret, de trouver un théorème qui “s’applique presque”.... Le rôle des contreexemples est justement de délimiter le possible, et ce n’est pas par perversité (ou en tout cas pas totalement) que les textes mathématiques exhibent des monstres. M. DEMAZURE, 1987 Abstract. Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the 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