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49
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Sharp estimates for the arithmetic Nullstellensatz
 Duke Math. J
"... We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which ..."
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Cited by 25 (2 self)
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We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine
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.
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 21 (13 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
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 20 (7 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.
Sharp estimates for the arithmetic
 Nullstellensatz, Duke Math. J. 109
, 2001
"... Nullstellensatz ..."
The Complexity of Quantifier Elimination and Cylindrical Algebraic Decomposition
 Proceedings ISSAC 2007
, 2007
"... This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simp ..."
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Cited by 18 (8 self)
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This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction — roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worstcase running times of several modern quantifier elimination algorithms and CADbased quantifier elimination when the number of quantifier alternations is constant.
Betti numbers of semialgebraic sets defined by quantifierfree formulae
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Let X be a semialgebraic set in R n defined by a Boolean combination of atomic formulae of the kind h ∗ 0where∗∈{>, ≥,=}, deg(h)
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Cited by 17 (4 self)
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Let X be a semialgebraic set in R n defined by a Boolean combination of atomic formulae of the kind h ∗ 0where∗∈{>, ≥,=}, deg(h) <d, and the number of distinct polynomials h is k. We prove that the sum of Betti numbers of X is less than O(k 2 d) n.
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.
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.