Connectedness in Tropical Combinatorics

Essentials of Tropical Combinatorics – Monograph

The tropical semi-ring is the triplet (R,⊕, ) with x⊕ y := min(x,y) and x y := x+ y . Its algebraic properties are somewhat modest, if one compares it with rings and fields, which are more common in most areas of mathematics. Still the distributive laws hold, and as an additional catch the addition is idempotent, that is, x⊕ x = x holds for all x ∈ R. As defined our tropical semi-ring is lackin...

Combinatorics of Tropical Hurwitz Cycles

We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as div...

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 [33] who considered the special case k = n− 1 and where all ...

Arithmetics and Combinatorics of Tropical Severi Varieties of Univariate Polynomials

We give a description of the tropical Severi variety of univariate polynomials of degree n having two double roots. We show that, as a set, it is given as the union of three explicit types of cones of maximal dimension n − 1, where only cones of two of these types are cones of the secondary fan of {0, . . . , n}. Through Kapranov’s theorem, this goal is achieved by a careful study of the possib...

Combinatorics Enumerative Combinatorics 05