Proceedings of the third conference on Computers and mathematics
Semi-algebraic complexity of quotients and sign determination of remainders
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
MAPC: a library for efficient and exact manipulation of algebraic points and curves
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually!
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
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 and open curved kernel
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On the exact computation of the topology of real algebraic curves
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
On the complexity of real solving bivariate systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Real Algebraic Numbers: Complexity Analysis and Experimentation
Reliable Implementation of Real Number Algorithms: Theory and Practice
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
On solving systems of bivariate polynomials
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
On the complexity of solving bivariate systems: the case of non-singular solutions
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computer-aided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on Sturm-Habicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced object-oriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the public-domain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches.