Asymptotic computations of tropical refined invariants in genus 0 and 1

Abstract

Block and Göttsche introduced a Laurent polynomial multiplicity to count tropical curves. Itenberg and Mikhalkin then showed that this multiplicity leads to invariant counts called tropical refined invariants. Recently, Brugallé and Jaramillo-Puentes studied the polynomiality properties of the coefficients of these invariants and showed that for fixed genus g, the coefficients ultimately coincide with polynomials in the homology class of the curves we look at. We call the generating series of these polynomials asymptotic refined invariant. In genus 0, the asymptotic refined invariant has been computed by the second author in the h-transverse case. In this paper, we give a new proof of the formula for the asymptotic refined invariant for g = 0 using variations on the floor diagram algorithm. This technique allows also to compute the asymptotic refined invariant for g = 1. The result exhibits surprising regularity properties related to the generating series of partition numbers and quasi-modular forms.