Polynomiality properties of tropical refined invariants

TL;DR

This paper investigates the polynomial behavior of coefficients in tropical refined invariants of toric surfaces, revealing new polynomiality properties and extending results to include additional parameters, with implications for complex and real enumerative geometry.

Contribution

It proves polynomiality of small codegree coefficients in tropical refined invariants and extends these results to include parameters related to psi classes in rational curves.

Findings

Coefficients of small codegree are polynomial in the Newton polygon for fixed genus.

Polynomiality extends to parameters counting psi classes in rational curves.

Results suggest new phenomena in complex enumerative geometry and potential links to real invariants.

Abstract

Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and G\"ottsche, and further extended by G\"ottsche and Schroeter in the case of rational curves. In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a somehow surprising resurgence, in some sort of dual setting, of the so-called node polynomials and G\"ottsche conjecture. Our methods are entirely combinatorial, hence our results may suggest phenomenons in complex enumerative geometry that have not been studied yet. In the particular case of rational curves, we extend our polynomiality results by including the extra parameter $s$ recording the number of $\psi$ classes. Contrary to the polynomiality with respect to $ \Delta$, the one with respect to $s$ may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of G\"ottsche-Schroeter invariants.