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.

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Reinhard Steffens, Thorsten Theobald
• Keywords: Intersection Curve, Polytopes, SIAMDM 2010, Tropical