Bounds for positive roots of polynomials
Journal of Computational and Applied Mathematics
Quantifier elimination: Optimal solution for two classical examples
Journal of Symbolic Computation
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Quantifier elimination for real algebra—the cubic case
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Bounds for absolute positiveness of multivariate polynomials
Journal of Symbolic Computation
Recent advances on determining the number of real roots of parametric polynomials
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Sylvester—Habicht sequences and fast Cauchy index computation
Journal of Symbolic Computation
A Discipline of Programming
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Near-optimal parameterization of the intersection of quadrics
Proceedings of the nineteenth annual symposium on Computational geometry
Towards faster real algebraic numbers
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Complete, exact, and efficient computations with cubic curves
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Towards and open curved kernel
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
The predicates of the Apollonius diagram: algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
A package for exact kinetic data structures and sweepline algorithms
Computational Geometry: Theory and Applications
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On the complexity of real solving bivariate systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the complexity of real root isolation using continued fractions
Theoretical Computer Science
Using signature sequences to classify intersection curves of two quadrics
Computer Aided Geometric Design
Resultant-based methods for plane curves intersection problems
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Computing a rational in between
ACM Communications in Computer Algebra
Univariate Algebraic Kernel and Application to Arrangements
SEA '09 Proceedings of the 8th International Symposium on Experimental Algorithms
Experimental evaluation and cross-benchmarking of univariate real solvers
Proceedings of the 2009 conference on Symbolic numeric computation
An algebraic approach to continuous collision detection for ellipsoids
Computer Aided Geometric Design
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Based on precomputed Sturm-Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of the input polynomial coefficients. Although the roots are algebraic numbers and can be expressed by radicals, such representation involves some roots of complex numbers. This is inefficient, and hard to handle in applications in geometric computing and quantifier elimination. We also define rational isolating points between the roots of the quintic. We combine these results with a simple version of Rational Univariate Representation to isolate all common real roots of a bivariate system of rational polynomials of total degree @?2 and to compute the multiplicity of these roots. We present our software within library synaps and perform experiments and comparisons with several public-domain implementations. Our package is 2-10 times faster than numerical methods and exact subdivision-based methods, including software with intrinsic filtering.