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SIAMDM
2010

Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

13 years 10 months ago
Combinatorics and Genus of Tropical Intersections and Ehrhart Theory
Let g1, . . . , gk be tropical polynomials in n variables with Newton polytopes P1, . . . , Pk. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by g1, . . . , gk, such as the f-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland’s work [30] who 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 P1, . . . , Pk. 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.
Reinhard Steffens, Thorsten Theobald
Added 30 Jan 2011
Updated 30 Jan 2011
Type Journal
Year 2010
Where SIAMDM
Authors Reinhard Steffens, Thorsten Theobald
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