Tropical Intersections and Rational Functions on Graphs

This write-up is concerned with the intersections of tropical curves and their relationship to rational functions on graphs. It is based on projects for Bernd Sturmfel’s 255 course in Fall 2011, Martin Olsson’s 256B course in Spring 2012, and work with Matt Baker. Sections 2 and 3 run through previous results on tropical intersections in the transverse and non-transverse cases, respectively. Section 4 runs through various examples for the non-transverse case, and highlights the balancing conditions that arise. After offering a characterization of tropical curves in terms of weighted metric graphs and describing rational functions on them (sections 5 and 6), we present our main conjecture in section 7 on the relationship between intersection configurations and rational functions. We close with a treatment of a tropical Riemann-Roch theorem which may be of use in determining degrees of freedom for these intersections.