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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 18 (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.
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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Cited by 16 (0 self)
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.
A sharp bound for the CastelnuovoMumford regularity of subspace arrangements
"... Over the past twenty years rapid advances in computational algebraic geometry have generated increasing amounts of interest in quantifying the “complexity ” of ideals and modules. For a finitely generated module M over a polynomial ring S = k[x0,..., xn] with k an arbitrary field, we say that M is r ..."
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Cited by 11 (6 self)
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Over the past twenty years rapid advances in computational algebraic geometry have generated increasing amounts of interest in quantifying the “complexity ” of ideals and modules. For a finitely generated module M over a polynomial ring S = k[x0,..., xn] with k an arbitrary field, we say that M is rregular (in the
The Geometry in Constraint Logic Programs
 In Position Papers for the First Workshop on Principles and Practice of Constraint Programming
, 1993
"... Many applications of constraint programming languages concern geometric domains. We propose incorporating strong algorithmic techniques from the study of geometric and algebraic algorithms into the implementation of constraint programming languages. Interesting new computational problems in computat ..."
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Cited by 3 (1 self)
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Many applications of constraint programming languages concern geometric domains. We propose incorporating strong algorithmic techniques from the study of geometric and algebraic algorithms into the implementation of constraint programming languages. Interesting new computational problems in computational geometry and computer algebra arises from such considerations. We look at what is known and what needs to be addressed.
On Arithmetical Formulas Whose Jacobians are Gröbner Bases
, 2000
"... We exhibit classes of polynomials whose sets of kth partial derivatives form Gro bner bases for all k, with respect to all term orders. The classes are defined by syntactic constraints on arithmetical formulas defining the polynomials. Readonce formulas without constants have this property for ..."
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We exhibit classes of polynomials whose sets of kth partial derivatives form Gro bner bases for all k, with respect to all term orders. The classes are defined by syntactic constraints on arithmetical formulas defining the polynomials. Readonce formulas without constants have this property for all k, while those with constants have a weaker "Grobnerbounding" property introduced here. For k = 1 the same properties hold even with arbitrary powering of subterms of the formulas. 1 Introduction One of the great unsolved problems of computational mathematics is finding lower bounds on the size of arithmetical circuits or formulas defining certain polynomial functions over a given field F . A famous example concerns the determinant polynomials d n and permanent polynomials p n in F [x 1 ; : : : ; x n 2 ], defined to be the determinant (respectively, permanent) of the n \Theta n matrix M n of variables x 1 ; : : : ; x n 2 (in rowmajor order). It is known that the d n have arithme...
On Polynomial Ideals, Their Complexity, and Applications
, 1995
"... A polynomial ideal membership problem is a (w+1)tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known th ..."
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A polynomial ideal membership problem is a (w+1)tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete. We discuss
On Arithmetical Formulas Whose Jacobians are Gröbner Bases
"... We exhibit classes of polynomials whose sets of kth partial derivatives form Gro ¨ bner bases for all k, with respect to all term orders. The classes are defined by syntactic constraints on arithmetical formulas defining the polynomials. Readonce formulas without constants have this property for al ..."
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We exhibit classes of polynomials whose sets of kth partial derivatives form Gro ¨ bner bases for all k, with respect to all term orders. The classes are defined by syntactic constraints on arithmetical formulas defining the polynomials. Readonce formulas without constants have this property for all k, while those with constants have a weaker “Gröbnerbounding ” property introduced here. For k = 1 the same properties hold even with arbitrary powering of subterms of the formulas. 1
§1. Problem of Algebra Lecture 0 Page 1 Lecture 0
"... This lecture is an orientation on the central problems that concern us. Specifically, we identify three families of “Fundamental Problems ” in algorithmic algebra (§1 –§3). In the rest of the lecture (§4– §9), we briefly discuss the complexitytheoretic background. §10 collects some common mathemati ..."
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This lecture is an orientation on the central problems that concern us. Specifically, we identify three families of “Fundamental Problems ” in algorithmic algebra (§1 –§3). In the rest of the lecture (§4– §9), we briefly discuss the complexitytheoretic background. §10 collects some common mathematical terminology while §11 introduces computer algebra systems. The reader may prefer to skip §411