## triangulation

 swMATH ID: 10357 Software Authors: Emiris, Ioannis Z.; Fisikopoulos, Vissarion; Konaxis, Christos; Peñaranda, Luis Description: 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 Homepage: http://doc.cgal.org/latest/Triangulation_2/classCGAL_1_1Delaunay__triangulation__2.html Dependencies: CGAL Keywords: general dimension; convex hull; regular triangulation; secondary polytope; resultant; CGAL implementation; experimental complexity Related Software: respol; TOPCOM; CGAL; NumericalNP; Bertini.m2; Bertini; SageMath; Macaulay2; Resultants; Maple; Hull; Eigen; polymake; LinBox; iB4e; TropLi Cited in: 5 Publications
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### Cited by 7 Authors

 3 Emiris, Ioannis Z. 3 Fisikopoulos, Vissarion 2 Konaxis, Christos 2 Peñaranda, Luis Mariano 1 Brysiewicz, Taylor 1 Gärtner, Bernd 1 Laroche, Clément

### Cited in 5 Serials

 1 Computer Aided Geometric Design 1 Journal of Symbolic Computation 1 International Journal of Computational Geometry & Applications 1 Computational Geometry 1 Mathematics in Computer Science

### Cited in 4 Fields

 3 Numerical analysis (65-XX) 2 Computer science (68-XX) 1 Algebraic geometry (14-XX) 1 Convex and discrete geometry (52-XX)