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49
Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers
 J. Complexity
, 2000
"... We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set o ..."
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Cited by 60 (2 self)
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We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set of points obtained from the intersection of the component with a generic transverse affine subspace. Our algorithm is incremental in the number of equations to be solved. Its complexity is mainly cubic in the maximum of the degrees of the solution sets of the intermediate systems counting multiplicities. Our method is designed for coefficient fields having characteristic zero or big enough with respect to the number of solutions. If the base field is the field of the rational numbers then the resolution is first performed modulo a random prime number after we have applied a random change of coordinates. Then we search for coordinates with small integers and lift the solutions up to the rational numbers. Our implementation is available within our package Kronecker from version 0.166, which is written in the Magma computer algebra system. 1
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 29 (19 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.
Computing Parametric Geometric Resolutions
, 2001
"... Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we f ..."
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Cited by 29 (8 self)
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Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 19 (9 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Differential equations for algebraic functions
 ISSAC’07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
, 2007
"... Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose deg ..."
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Cited by 16 (6 self)
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Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series. 1.
Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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Fast algorithms for Taylor shifts and certain difference equations (Extended Abstract)
, 1997
"... We analyze six algorithms for computing integral Taylor shifts for polynomials with integral coefficients. We present and analyze a new algorithm for solving the "key equation" which occurs in many rational and hypergeometric summation algorithms. In a special case, our algorithm is asympt ..."
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Cited by 11 (1 self)
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We analyze six algorithms for computing integral Taylor shifts for polynomials with integral coefficients. We present and analyze a new algorithm for solving the "key equation" which occurs in many rational and hypergeometric summation algorithms. In a special case, our algorithm is asymptotically faster than previously known methods. We give experimental results for our algorithms.
Deformation techniques for sparse systems
 A POISSON FORMULA FOR THE SPARSE RESULTANT 33
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Matrix Rank Certification
, 2001
"... Randomized algorithms are given for computing the rank of a matrix over a field of... ..."
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Cited by 9 (1 self)
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Randomized algorithms are given for computing the rank of a matrix over a field of...
Solving Linear Systems of Equations with Randomization
 Augmentation and Aggregation, Linear Algebra and Its Applications
"... Seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization, augmentation, and aggregation to reduce our task to computations with well conditioned matrices of full rank. Our algorithms avoid pivoting and orthogonalization, preser ..."
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Cited by 9 (8 self)
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Seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization, augmentation, and aggregation to reduce our task to computations with well conditioned matrices of full rank. Our algorithms avoid pivoting and orthogonalization, preserve matrix structure and sparseness, and in the case of an ill conditioned input perform only a small part of the computations with high accuracy. We extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities. Our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned. Our study can be of independent technical interest: we extend the known results on conditioning of random matrices to randomized preconditioning, estimate the condition numbers of randomly augmented matrices, and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations. AMS Classification: