A polynomial-time algorithm for the topological type of a real algebraic curve
Journal of Symbolic Computation
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On the worst-case arithmetic complexity of approximating zeros of systems of polynomials
SIAM Journal on Computing
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Singular points of algebraic curves
Journal of Symbolic Computation
Proceedings of the third conference on Computers and mathematics
Binary segmentation for multivariate polynomials
Journal of Complexity
An improved upper complexity bound for the topology computation of a real algebraic plane curve
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Asymptotically fast computation of subresultants
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
New structure theorem for subresultants
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
Solving projective complete intersection faster
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Sylvester—Habicht sequences and fast Cauchy index computation
Journal of Symbolic Computation
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Theoretical Computer Science - Latin American theoretical informatics
A modular GCD algorithm over number fields presented with multiple extensions
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
On the exact computation of the topology of real algebraic curves
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
The implicit structure of ridges of a smooth parametric surface
Computer Aided Geometric Design
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Real solving of bivariate polynomial systems
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Exact geometric-topological analysis of algebraic surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Real algebraic numbers and polynomial systems of small degree
Theoretical Computer Science
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Root isolation for bivariate polynomial systems with local generic position method
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
An efficient algorithm for the stratification and triangulation of an algebraic surface
Computational Geometry: Theory and Applications
On the Complexity of Reliable Root Approximation
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
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We consider exact real solving of well-constrained, bivariate systems of relatively prime polynomials. The main problem is to compute all common real roots in isolating interval representation, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of ÕB(N14) for the purely projection-based method, and ÕB(N12) for two subresultants-based methods: these ignore polylogarithmic factors, and N bounds the degree and the bitsize of the polynomials. The previous record bound was ÕB(N14). Our main tool is signed subresultant sequences, extended to several variables by binary segmentation. We exploit advances on the complexity of univariate root isolation, and extend them to multipoint sign evaluation, sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in. All algorithms have been implemented in maple, in conjunction with numeric filtering. We compare them against fgb/rs and synaps; we also consider maple libraries insulate and top, which compute curve topology. Our software is among the most robust, and its runtimes are within a small constant factor, with respect to the C/C++ libraries.