A Riemann-roch Theorem in Tropical Geometry

Recently, Baker and Norine have proven a Riemann-Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann-Roch theorem for divisors on (abstract) tropical curves. Tropical algebraic geometry is a recent branch of mathematics that establishes deep relations between algebro-geometric and purely combinatorial objects. Ideally, every construction and theorem of algebraic geometry should have a tropical (i.e. combinatorial) counterpart that is then hopefully easier to understand — e.g. the tropical counterpart of n-dimensional varieties are certain n-dimensional polyhedral complexes. In this paper we will establish a tropical counterpart of the well-known Riemann-Roch theorem for divisors on curves. Let us briefly describe the idea of our result. Following Mikhalkin, an (abstract) tropical curve is simply a connected metric graph Γ. A rational function on Γ is a continuous, piecewise linear real-valued function f with integer slopes. For such a function and any point P ∈ Γ the order ordP f of f in P is the sum of the slopes of f for all edges emanating from P. For example, the following picture shows a rational function f on a tropical curve Γ with simple zeroes at P2 and P5 (i.e. ordP2 f = ordP5 f = 1), and simple poles at P3 and P4 (i.e. ordP3 f = ordP4 f = −1).