Universal polynomials for tropical refined invariants in genus 0

TL;DR

This paper proves the existence of universal polynomials for tropical refined invariants in genus 0, providing explicit formulas and leveraging floor diagram techniques for rational enumeration.

Contribution

It establishes the existence of universal polynomials depending only on genus and codegree for tropical refined invariants, with explicit formulas for genus 0.

Findings

Universal polynomials exist for rational enumeration in genus 0.

Explicit formulas for these polynomials are provided.

Floor diagrams are used as a key proof technique.

Abstract

In 2022, Brugall{\'e} and Jaramillo-Puentes showed that the coefficients of small codegree of the tropical refined invariant are polynomial in the Newton polygon. This raised the question of the existence of universal polynomials giving these coefficients, i.e. polynomials depending only on the genus and the codegree, and with variables the combinatorial data of the Newton polygon.In this paper we show that such universal polynomials exist for rational enumeration, and we give an explicit formula. The proof relies on the manipulation of floor diagrams.