Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points where a linear or quadratic function is non-negative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-ofthe-art implementations, that is, Lgp and Maple's Isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve.Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry.We have implemented our algorithms as prototypical contributions to the C++-project Cgal. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple's Isolate for polynomial system solving, and Cgal's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones. (Michael Sagraloff) 1 G C is isotopic to C if there exists a continuous mapping φ :Outline. The bivariate solver Bisolve is discussed in Section 2. In Section 3, we introduce GeoTop to analyze a single algebraic curve. The latter section particularly features two parts, that is, the presentation of a complete method Lift-BS in Section 3.2.1 that is based on Bisolve, and the presentation of the symbolic-numeric method Lift-NT in Section 3.2.2. Lift-NT uses a numerical solver whose details are given in Appendix A. Bisolve and GeoTop are finally utilized in Section 4 in order to enable the computation of arrangements of algebraic curves. The presented algorithms allow speedups, among other things, due to the use of graphics hardware for symbolic operations as described in Section 5. Our algorithms are prototypically implemented in the Cgal project. Section 6 gives necessary details and also features many experiments that show the performance of the new approach. We conclude in Section 7 and outline further directions of research.

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