Tropical Invariants from the Secondary Fan

In this paper, we consider weighted counts of tropical plane curves of particular combinatorial type through a certain number of generic points. We give a criterion, derived from tropical intersection theory on the secondary fan, for a weighted count to give a number invariant of the position of the points. By computing a certain intersection multiplicity, we show how Mikhalkin's approach to computing Gromov-Witten invariants fits into our approach. This begins to address a question raised by Dickenstein, Feichtner, and Sturmfels. We also give a geometric interpretation of the numbers we produce involving Chow quotients, and provide a counterexample showing that the tropical Severi variety is not always supported on the secondary fan. This paper is a revision of the preprint, "The Tropical Degree of Cones in the Secondary Fan."