An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from $18$ to $100$ times. We implement our algorithm using the experimental CGAL package {\tt triangulation}. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90\% and 105\% of the true volume, runtime is reduced up to $25$ times. Our method computes instances of $5$-, $6$- or $7$-dimensional polytopes with $35$K, $23$K or $500$ vertices, resp., within $2$hr. Compared to tropical geometry software, ours is faster up to dimension $5$ or $6$, and competitive in higher dimensions

Gemischte Volumina, gemischte Ehrhart-Theorie und deren Anwendungen in tropischer Geometry und Gestaengekonfigurationsproblemen

The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer points in Minkowski sums, new expressions in terms of mixed volumes of combinatorial quantities in tropical geometry and furthermore we employ mixed volume techniques to obtain bounds in certain graph embedding problems.Ziel dieser Arbeit ist die Diskussion gemischter Volumina, ihres Zusammenspiels mit der algebraischen Geometrie, der diskreten Geometrie und der tropischen Geometrie sowie deren Anwendungen im Bereich von Gestaenge-Konfigurationsproblemen. Wir praesentieren insbesondere neue Methoden zur Berechnung gemischter Volumina, einen neuen Zugang zur Ehrhart Theorie, welcher gemischte Volumina mit der Enumeration ganzzahliger Punkte in Minkowski-Summen verbindet, neue Formeln, die kombinatorische Groessen der tropischen Geometrie mithilfe gemischter Volumina beschreiben, und einen neuen Ansatz zur Verwendung gemischter Volumina zur Loesung eines Einbettungsproblems der Graphentheorie

Decomposing The Secondary Cayley Polytope

The vertices of the secondary polytope of a point configuration correspond to its regular triangulations. The Cayley trick links triangulations of one point configuration, called the Cayley polytope, to the fine mixed subdivisions of a tuple of point configurations. In this paper we investigate the secondary polytope of this Cayley polytope. Its vertices correspond to all regular mixed subdivisions of a tuple of point configurations. We demonstrate that it equals the Minkowski sum of polytopes, which we call mixed secondary polytopes, whose vertices correspond to regular-cell configurations