Computing tropical bitangents to smooth quartic curves in polymake

TL;DR

This paper introduces a polymake extension and database for smooth tropical quartic curves, providing algorithms to analyze tropical bitangents and their real lifting conditions, leading to a tropical proof of classical bitangent counts.

Contribution

The paper presents new computational tools and algorithms for analyzing tropical quartic curves and their bitangents, enabling a tropical proof of classical algebraic geometry results.

Findings

Development of the TropicalQuarticCurves extension in polymake

Implementation of algorithms for tropical bitangent analysis

A tropical proof of Plücker and Zeuthen's count of real bitangents

Abstract

In this article we introduce the recently developed polymake extension TropicalQuarticCurves and its associated database entry in polyDB dealing with smooth tropical quartic curves. We report on algorithms implemented to analyze tropical bitangents and their lifting conditions over real closed valued fields. The new functions and data were used by the authors to provide a tropical proof of Pl\"ucker and Zeuthen's count of real bitangents to smooth quartic curves.