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 α - n - 1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per 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.
Highlights
We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction
By Cayley’s trick (Proposition 2) the regular tight mixed subdivisions of the Minkowski sum A0 + · · · + An are in bijection with the regular triangulations of A, which are in bijection with the vertices of the secondary polytope Σ(A)
The resultant is the most fundamental tool in elimination theory, it is instrumental in system solving and optimization, and is crucial in geometric modeling, most notably for changing the representation of parametric hypersurfaces to implicit
Summary
We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. In the critical step of computing the convex hull of the resultant polytope, uses triangulation. Our algorithm computes its Newton polytope with vertices (0, 2, 0, 1, 1), (0, 0, 2, 2, 0), (2, 0, 0, 0, 2); it contains 4 lattice points, corresponding to 4 potential resultant monomials a21 b1 b0 , a20 b21 , a2 a0 b1 b0 , a22 b20 .
Sign-in/Register to view highlights & summary 
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call