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2 results on '"PUISEUX SERIES FIELD"'

1. Tropicalization of facets of polytopes.

Author

Allamigeon, Xavier and Katz, Ricardo D.

Subjects
*POLYTOPES, *MATHEMATICAL series, *INTERSECTION theory, *REPRESENTATION theory, *CONVEX domains
Abstract

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu. [ABSTRACT FROM AUTHOR]

Published
2017
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2. Tropicalization of facets of polytopes

Author

Xavier Allamigeon, Ricardo D. Katz, TROPICAL (TROPICAL), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Centro Franco Argentino de Ciencias de la Información y de Sistemas [Rosario] (CIFASIS), Universidad Nacional de Rosario [Santa Fe]-Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET), The authors were partially supported by the PGMO program of EDF and Fondation Mathématique Jacques Hadamard. An important part of this work was done during a visit of Ricardo D. Katz to the Ecole Polytechnique, which was funded by Digitéo. Ricardo D. Katz was also partially supported by CONICET Grant PIP 11220110101026., and Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET)-Universidad Nacional de Rosario [Santa Fe]

Subjects
TROPICAL CONVEXITY, Matemáticas, 14T05, 52A01, FACET-DEFINING HALF-SPACES, Polytope, 0102 computer and information sciences, PUISEUX SERIES FIELD, 01 natural sciences, Puiseux series, Matemática Pura, Combinatorics, Lift (mathematics), purl.org/becyt/ford/1 [https], HAHN SERIES FIELD, Tropical convexity, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics::Metric Geometry, Mathematics - Combinatorics, Puiseux series field, 0101 mathematics, Physics::Atmospheric and Oceanic Physics, Mathematics, Numerical Analysis, Algebra and Number Theory, Conjecture, Mathematics::Combinatorics, 010102 general mathematics, purl.org/becyt/ford/1.1 [https], external representations, 14T05 (Primary), 52A01 (Secondary), facet-defining half-spaces, 010201 computation theory & mathematics, EXTERNAL REPRESENTATIONS, polytopes, Combinatorics (math.CO), Geometry and Topology, General position, POLYTOPES, CIENCIAS NATURALES Y EXACTAS
Abstract

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu., 18 pages, 6 figures; v2: major revision

Published
2014
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