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46
On triangular decompositions of algebraic varieties
 Presented at the MEGA2000 Conference
, 1999
"... We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifti ..."
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Cited by 68 (37 self)
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We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifting property for calculations modulo regular chains. This allows a good management of the intermediate computations, as confirmed by several implementations and applications of this work. Our algorithm is also well suited for parallel execution.
Polar Varieties and Computation of one Point in each Connected Component of a Smooth Real Algebraic Set
, 2003
"... Let f1,..., fs be polynomials in Q[X1,..., Xn] that generate a radical ideal and let V be their complex zeroset. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each ..."
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Cited by 33 (16 self)
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Let f1,..., fs be polynomials in Q[X1,..., Xn] that generate a radical ideal and let V be their complex zeroset. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each connected component of V ∩ R n. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to noncompact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity.
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.
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Cited by 16 (3 self)
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.
Change of ordering for regular chains in positive dimension
 IN ILIAS S. KOTSIREAS, EDITOR, MAPLE CONFERENCE 2006
, 2006
"... We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to d ..."
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Cited by 16 (7 self)
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We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to dimension zero and using NewtonHensel lifting techniques. The problems raised by the choice of the specialization points, the lack of the (crucial) information of what are the free and algebraic variables for the new ordering, and the efficiency regarding the other methods are discussed. Strong hypotheses (but not unusual) for the initial regular chain are required. Change of ordering in dimension zero is taken as a subroutine.
Improved dense multivariate polynomial factorization algorithms
 J. Symbolic Comput
, 2005
"... We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the numb ..."
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Cited by 15 (3 self)
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We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem. Key words: Polynomial factorization, Hensel lifting, Bertini’s irreducibility theorem.
Algorithms for Computing Triangular Decomposition of Polynomial Systems
, 2011
"... We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we reg ..."
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Cited by 15 (13 self)
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We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.
Testing sign conditions on a multivariate polynomial and applications
 MATHEMATICS IN COMPUTER SCIENCE
"... Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected c ..."
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Cited by 13 (6 self)
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Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f − e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set K0(f) of critical values of the mapping f and K∞(f) of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semialgebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach.
High Probability Analysis of the Condition Number of Sparse Polynomial Systems
 THEORETICAL COMPUTER SCIENCE, SPECIAL ISSUE ON ALGEBRAIC AND NUMERICAL ALGORITHMS
, 2002
"... Let F:=(f1,...,fn) be a random polynomial system with fixed ntuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/ε. The bound depends on an integral of a differential form on a toric manifold and admits a simple expli ..."
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Cited by 13 (8 self)
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Let F:=(f1,...,fn) be a random polynomial system with fixed ntuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/ε. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying covariances) are all identical. We also consider polynomials with real coefficients and give bounds for the expected number of real roots and (restricted) condition number. Using a Kähler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form, thus recovering the classical mixed volume when U = (C ∗ ) n.
Why polyhedra matter in nonlinear equation solving
, 2003
"... We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the fol ..."
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Cited by 12 (4 self)
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We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely selfcontained proof of an extension of Bernstein’s Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area — a quantity intimately related to counting complex roots in the plane.