A Tropical Computation of Refined Toric Invariants

TL;DR

This paper extends the computation of refined tropical invariants for real rational curves in toric surfaces to configurations with conjugated point pairs on the same boundary component, building on Mikhalkin's earlier work.

Contribution

It generalizes the tropical computation of refined invariants to include conjugated point pairs on a single boundary component, expanding previous real point configurations.

Findings

Computed refined invariants for configurations with conjugated pairs on the same boundary component.

Extended tropical geometry methods to more general point configurations.

Confirmed invariance of counts under new conjugation conditions.

Abstract

In arXiv:1505.04338(4), G. Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through certain conjugation invariant set of points on the toric boundary of the surface. Such a set consists of real points and pairs of complex conjugated points. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, he managed to compute these invariants if all the points of the configuration are real. In this paper we address the case when the configuration contains some pairs of conjugated points, all belonging to the same component of the toric boundary.