Tropical complete intersection curves

A tropical complete intersection curve C ⊆ Rn+1 is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected. 1 Notation and definitions We work over the tropical semifield Rtr= (R,⊕,⊙) = (R,max,+). A tropical (Laurent) polynomial in variables x1, . . . , xm is an expression of the form (1) f = ⊕ a=(a1,...,am)∈A λa x a1 1 · · ·x am m = max a∈A {λa + a1x1 + · · ·+ amxm}, where the support setA is a finite subset of Z, and the coefficients λa are real numbers. (In the middle expression of (1), all products and powers are tropical.) The convex hull of A in R is called the Newton polytope of f , denoted ∆f . Any tropical polynomial f induces a regular lattice subdivision of ∆f in the following way: With f as in (1), let the lifted Newton polytope ∆̃f be the polyhedron defined as ∆̃f := conv({(a, t) | a ∈ A, t ≤ λa}) ⊆ ∆f ×R ⊆ R m ×R Furthermore, we define the top complex Tf to be the complex whose maximal cells are the bounded facets of ∆̃f . Projecting the cells of Tf to R m by deleting the last coordinate gives a collection of lattice polytopes contained in ∆f , forming a regular subdivision of ∆f . We denote this subdivision by Subdiv(f). The standard volume form on R is denoted by volm( · ), or simply vol( · ) if the space is clear from the context. 1.1 Tropical hypersurfaces Note that any tropical polynomial f(x1, . . . , xm) is a convex, piecewise linear function f : R → R. ∗Department of Mathematics, University of Oslo, Norway. Email : [email protected]