An oracle-based, output-sensitive algorithm for projections of resultant polytopes. 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 ℤ n . 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.
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References in zbMATH (referenced in 5 articles )
Showing results 1 to 5 of 5.
- Brysiewicz, Taylor: Numerical software to compute Newton polytopes and tropical membership (2020)
- Emiris, Ioannis Z.; Konaxis, Christos; Laroche, Clément: Implicit representations of high-codimension varieties (2019)
- Emiris, Ioannis Z.; Fisikopoulos, Vissarion; Gärtner, Bernd: Efficient edge-skeleton computation for polytopes defined by oracles (2016)
- Fisikopoulos, Vissarion; Peñaranda, Luis: Faster geometric algorithms via dynamic determinant computation (2016)
- Emiris, Ioannis Z.; Fisikopoulos, Vissarion; Konaxis, Christos; Peñaranda, Luis: An oracle-based, output-sensitive algorithm for projections of resultant polytopes (2013)