An Oraclebased, Outputsensitive Algorithm for Projections of Resultant Polytopes
Abstract
We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex and halfspacerepresentations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is alphan1, where the input consists of alpha points in Z^n. Our approach is outputsensitive as it makes one oracle call per vertex and facet. It extends to any polytope whose oraclebased definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5, 6 and 7dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in <1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.
 Publication:

arXiv eprints
 Pub Date:
 August 2011
 arXiv:
 arXiv:1108.5985
 Bibcode:
 2011arXiv1108.5985E
 Keywords:

 Computer Science  Symbolic Computation;
 Computer Science  Computational Geometry
 EPrint:
 27 pages, 7 figures, 4 tables. In IJCGA (invited papers from SoCG '12)