An Oracle-based, Output-sensitive Algorithm for Projections of Resultant Polytopes

TL;DR

This paper introduces an output-sensitive, oracle-based algorithm for efficiently computing the Newton polytope of the resultant and its projections, significantly improving performance over existing methods in high dimensions.

Contribution

The authors develop a novel oracle-based algorithm that computes resultants' polytopes efficiently, extending to other polytopes and outperforming tropical geometry software in higher dimensions.

Findings

Successfully computed high-dimensional polytopes within hours.

Achieved sub-second computation times for important surface equations.

Enhanced algorithm speed with hashing determinantal predicates by up to 100 times.

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 halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is alpha-n-1, where the input consists of alpha points in Z^n. Our approach is output-sensitive as it makes one oracle call per vertex and facet. It extends to any polytope whose oracle-based 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 7-dimensional 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.