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We present a novel certified and complete algorithm to compute arrangementsof real planar algebraic curves. It provides a geometric-topological analysisof the decomposition of the plane induced by a finite number of algebraiccurves in terms of a cylindrical algebraic decomposition. From a high-levelperspective, the overall method splits into two main subroutines, namely analgorithm denoted Bisolve to isolate the real solutions of a zero-dimensionalbivariate system, and an algorithm denoted GeoTop to analyze a single algebraiccurve.  Compared to existing approaches based on elimination techniques, weconsiderably improve the corresponding lifting steps in both subroutines. As aresult, generic position of the input system is never assumed, and thus ouralgorithm never demands for any change of coordinates. In addition, wesignificantly limit the types of involved exact operations, that is, we onlyuse resultant and gcd computations as purely symbolic operations. The latterresults 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 theC++-project CGAL. They exploit graphics hardware to expedite the symboliccomputations. We have also compared our implementation with the currentreference implementations, that is, LGP and Maple's Isolate for polynomialsystem solving, and CGAL's bivariate algebraic kernel for analyses andarrangement computations of algebraic curves. For various series of challenginginstances, our exhaustive experiments show that the new implementationsoutperform the existing ones.

Exact Symbolic-Numeric Computation of Planar Algebraic Curves