An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

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-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Second order surface analysis using hybrid symbolic and numeric operators

Journal ArticleResults from analyzing the curvature of a surface can be used to improve the implementation, efficiency, and effectiveness of manufacturing and visualization of sculptured surfaces. In this paper, we develop a robust method using hybrid symbolic and numeric operators to create trimmed surfaces each of which is solely convex, concave, or saddle and partitions the original surface. The same method is also used to identify regions whose curvature lies within prespecified bounds