Four results on randomized incremental constructions
Computational Geometry: Theory and Applications
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
How good are convex hull algorithms?
Computational Geometry: Theory and Applications
Sign determination in residue number systems
Theoretical Computer Science - Special issue on real numbers and computers
Polymake: an approach to modular software design in computational geometry
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Incremental Convex Hull Algorithms Are Not Output Sensitive
ISAAC '96 Proceedings of the 7th International Symposium on Algorithms and Computation
On the complexity of computing determinants
Computational Complexity
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension
Proceedings of the twenty-fifth annual symposium on Computational geometry
Triangulations: Structures for Algorithms and Applications
Triangulations: Structures for Algorithms and Applications
Implicitization of curves and surfaces using predicted support
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Faster geometric algorithms via dynamic determinant computation
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Sparse implicitization using support prediction
ACM Communications in Computer Algebra
Implicitization of curves and (hyper)surfaces using predicted support
Theoretical Computer Science
Combinatorics of 4-dimensional resultant polytopes
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
| Hi-index | 0.00 |
We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6- or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.