arXiv subjects

Michael Sagraloff

Publications and source records attributed to Michael Sagraloff.

21 records · Page 2Linked to original sources

arXiv2011

Arrangement Computation for Planar Algebraic Curves

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm 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 of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library CGAL. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, CGAL's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

cs.SC↗

arXiv2010

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.

cs.MS↗

arXiv2006

Special Linear Series and Syzygies of Canonical Curves of Genus 9

In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves $C$ of genus 9. This includes a collection of all possible Betti tables for $C$. Moreover a direct correspondence between these Betti tables and the number and types of special linear series on $C$ is given. Especially for $\operatorname{Cliff}(C)=3$ the curve $C$ is contained in determinantal surface $Y$ on a 4-dimensional rational normal scroll $X\subset \mathbb{P}^8$ constructed from a base point free pencil of divisors of degree 5.

math.AG↗