Results 1  10
of
22
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 90 (17 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
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 31 (0 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
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 28 (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 26 (17 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.
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 18 (10 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.
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 13 (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.
Kronecker’s and Newton’s approaches to solving: A First Comparison
, 1999
"... In these pages we make a first attempt to compute efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker’s solution (from the symbolic approach) with approximate zero theory (introduced by M. Shub & S ..."
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Cited by 13 (5 self)
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In these pages we make a first attempt to compute efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker’s solution (from the symbolic approach) with approximate zero theory (introduced by M. Shub & S. Smale as a foundation of numerical analysis). To this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker’s solution into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting fields and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.
Algebraic Geometry Over Four Rings and the Frontier to Tractability
 CONTEMPORARY MATHEMATICS
"... We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algeb ..."
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Cited by 8 (4 self)
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We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algebraic set over C (c) the number of connected components of a semialgebraic set We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes 989 and 9989, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper
Estimates on the distribution of the condition number of singular matrices
 Found. Comput. Math
"... We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional ..."
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Cited by 6 (4 self)
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We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equi–dimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.
Dedekind Zeta Functions and the Complexity of Hilbert’s Nullstellensatz
, 2008
"... Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the ..."
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Cited by 5 (4 self)
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Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication HN ̸∈P = ⇒ P ̸=NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.