Asymptotic computations of tropical refined invariants in genus 0 and 1
Thomas Blomme, Gurvan Mével · Journal de l’École polytechnique — Mathématiques · 2025
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 that we consider. 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 also enables us 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.