NDLI: Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Steffens, Reinhard Theobald, Thorsten
Copyright Year	2010
Abstract	Let $g_1,\dots,g_k$ be tropical polynomials in n variables with Newton polytopes $P_1,\dots,\allowbreak P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1,\dots,g_k$, such as the f-vector, the number of unbounded faces, and (in the case of a curve) the genus. Our point of departure is Vigeland's work [Tropical complete intersection curves, preprint, arXiv:math/0711.1962, 2007] which considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary k and arbitrary Newton polytopes $P_1,\dots,P_k$. This provides new formulas for the number of faces and the genus in terms of mixed volumes. By establishing some aspects of a mixed version of Ehrhart theory we show that the genus of a tropical intersection curve equals the genus of a toric intersection curve corresponding to the same Newton polytopes.
Starting Page	17
Ending Page	32
Page Count	16
File Format	PDF
ISSN	08954801
DOI	10.1137/09075024X
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	24
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-02-05
Access Restriction	Subscribed
Subject Keyword	Lattice polytopes genus Ehrhart theory Tropical geometry tropical intersection tropical geometry
Content Type	Text
Resource Type	Article
Subject	Mathematics

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