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57
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 109 (19 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
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both meth ..."
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Cited by 44 (26 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities (SI) and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some experimentations on various data sets.
Almost tight recursion tree bounds for the Descartes method
 In Proc. Int. Symp. on Symbolic and Algebraic Computation
, 2006
"... We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound ..."
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Cited by 42 (3 self)
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We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients ai  < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.
On the asymptotic and practical complexity of solving bivariate systems over the reals
 JSC
"... This paper is concerned with exact real solving of wellconstrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexit ..."
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Cited by 23 (3 self)
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This paper is concerned with exact real solving of wellconstrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of eOB(N 14) for the purely projectionbased method, and eOB(N 12) for two subresultantbased methods: this notation ignores polylogarithmic factors, where N bounds the degree, and the bitsize of the polynomials. The previous record bound was eOB(N 14). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in eOB(N 12), whereas the previous bound was eOB(N 14). All algorithms have been implemented in maple, in conjunction with numeric filtering. We compare them against fgb/rs, system solvers from synaps, and maple libraries insulate and top, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries.
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 22 (10 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.
When Newton meets Descartes: A simple and fast algorithm to isolate the real roots of a polynomial
 CoRR
"... We introduce a novel algorithm denoted NEWDSC to isolate the real roots of a univariate squarefree polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f and performs exact operations on the coefficients of f i ..."
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Cited by 18 (5 self)
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We introduce a novel algorithm denoted NEWDSC to isolate the real roots of a univariate squarefree polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f and performs exact operations on the coefficients of f in each step. For the subdivision strategy, we combine Descartes ’ Rule of Signs and Newton iteration. More precisely, instead of using a fixed subdivision strategy such as bisection in each iteration, a Newton step based on the number of sign variations for an actual interval is considered, and, only if the Newton step fails, we fall back to bisection. Following this approach, our analysis shows that, for most iterations, quadratic convergence towards the real roots is achieved. In terms of complexity, our method induces a recursion tree of almost optimal size O(n · log(nτ)), where n denotes the degree of the polynomial and τ the bitsize of its coefficients. The latter bound constitutes an improvement by a factor of τ upon all existing subdivision methods for the task of isolating the real roots. In addition, we provide a bit complexity analysis showing that NEWDSC needs only Õ(n3τ) bit operations1 to isolate all real roots of f. This matches the best bound known for this fundamental problem. However, in comparison to the significantly more involved numerical algorithms by V. Pan and A. Schönhage, which achieve the same bit complexity for the task of isolating all complex roots, NEWDSC focuses on real root isolation, is much easier to access and to implement. 1.
Extremal real algebraic geometry and Adiscriminants
 Moscow Mathematical Journal
, 2007
"... We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We ..."
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Cited by 14 (6 self)
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We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the Adiscriminant, and give new bounds on the topology of certain Adiscriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations. 1
On the topology of real algebraic plane curves
, 2010
"... We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coo ..."
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Cited by 13 (2 self)
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We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use subresultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in nongeneric positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on highdegree nongeneric curves.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 13 (2 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm.