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Analysis of Real Algebraic Plane Curves


Kerber,  Michael
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Kerber, M. (2006). Analysis of Real Algebraic Plane Curves. Master Thesis, Universität des Saarlandes, Saarbrücken.

This work describes a new method to compute geometric properties of a real algebraic plane curve of arbitrary degree. These properties contain the topology of the curve as well as the location of singular points and vertical asymptotes. The algorithm is based on the Bitstream Descartes method (Eigenwillig et al.: "A Descartes Algorithm for Polynomials with Bit-Stream Coefficients", LNCS~3718), which computes exact information about the real roots of a polynomial from approximate coefficients. For symbolic calculations with algebraic numbers, especially for counting distinct real roots, it uses Sturm-Habicht sequences (Gonzalez-Vega et al.: "Sturm-Habicht Sequences \ldots", in: Caviness, Johnson(eds.): {\it Quantifier Elimination\ldots}, Springer, 1998), which are related to polynomial remainder sequences. Our work explains how to combine these methods to reduce the amount of symbolic calculations without losing exactness.\par The geometry of the curve is computed with respect to the predetermined coordinate system. The algorithm changes coordinates in some situations to bring the curve into a generic position, but a new technique transports the computed information back into the original system efficiently. The conditions for a generic position of the curve are less restrictive than in other approaches and can be checked more efficiently during the analysis.\par The algorithm has been implemented as part of the software library EXACUS. This work presents comprehensive experimental results. They show that the new approach consistently outperforms the method by Seidel and Wolpert ("On the Exact Computation \ldots", SCG 2005, 107--115) and the frequently cited algorithm of Gonzalez-Vega and Necula ("Efficient Topology Determination \ldots", {\it Comp.\ Aided Design} {\bf 19} (2002) 719--743). We therefore claim that our algorithm reflects the state-of-the-art in the resultant-based analysis of algebraic curves.