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Real Tropical Quartics and their Bitangents: Counting with Patchworking

Alheydis Geiger

TL;DR

The paper investigates the lifting of real tropical quartic bitangents, including non-generic cases, using Viro-style patchworking on real tropical curves $(\Gamma,\delta)$ with dual subdivisions of $4\Delta_2$ and sign vectors. It extends previous results to non-generic loci, showing that all real lifts for a given combinatorial type have the expected total-realness properties, with at most one non-totally real bitangent arising from the shape (C) in non-generic settings. A comprehensive computational map is produced, counting real combinatorial types by the number of real ovals and real bitangents (4,8,16,28) and making this data publicly available via polyDB, alongside an updated polymake extension. The work thus provides a complete, reproducible framework linking tropical dual subdivisions, patchworking, and real bitangent counts in plane quartics, with practical impact for real algebraic geometry and tropical geometry databases.

Abstract

For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of the combinatorial types of real tropical quartic curves according to their real topology and number of real bitangents. This paper is accompanied by an extension for polymake; the computational data is available in the database collection polyDB.

Real Tropical Quartics and their Bitangents: Counting with Patchworking

TL;DR

The paper investigates the lifting of real tropical quartic bitangents, including non-generic cases, using Viro-style patchworking on real tropical curves with dual subdivisions of and sign vectors. It extends previous results to non-generic loci, showing that all real lifts for a given combinatorial type have the expected total-realness properties, with at most one non-totally real bitangent arising from the shape (C) in non-generic settings. A comprehensive computational map is produced, counting real combinatorial types by the number of real ovals and real bitangents (4,8,16,28) and making this data publicly available via polyDB, alongside an updated polymake extension. The work thus provides a complete, reproducible framework linking tropical dual subdivisions, patchworking, and real bitangent counts in plane quartics, with practical impact for real algebraic geometry and tropical geometry databases.

Abstract

For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of the combinatorial types of real tropical quartic curves according to their real topology and number of real bitangents. This paper is accompanied by an extension for polymake; the computational data is available in the database collection polyDB.

Paper Structure

This paper contains 6 sections, 15 theorems, 6 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Counting Bitangents of smooth tropical quartics
  4. Data and Implementation
  5. polymake extension.
  6. Database polyDB.

Key Result

Theorem A

The sign conditions determined in CueMa20 are valid for all smooth tropical quartics, even if they are non-generic w.r.t. def:generic(i). Further, for a generic tropical quartic all real bitangents are totally real, while for smooth tropical quartics non-generic w.r.t. def:generic(i) at most one rea

Figures (6)

  • Figure 1: Illustration of the patchworking technique.
  • Figure 2: Amoeba for a part of a tropical plane curve showing the difference between a twisted and a not twisted edge.
  • Figure 3: A smooth tropical quartic curve with its 7 bitangent classes. The bitangent class of shape (C) is depicted as a red circle filled with white. The corresponding tropical bitangent line is depicted in blue.
  • Figure 4: Any tropical quartic with dual subdivision containing this partial subdivision (up to $S_3$) is never generic.
  • Figure 5: The Honeycomb subdivision on the right is a generic subdivision. The tropical curve on the left is dual to this subdivision, but the curve does not satisfy genericity condition \ref{['def:generic']}(i).
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem A: Theorems \ref{['thm:lift1']}, \ref{['thm:lift']}, Proposition \ref{['prop:totallyreal']}
  • Theorem B: Theorem \ref{['thm:main']}
  • Definition 2.1
  • Definition 2.2: texier2021topologyrealalgebraiccurves
  • Theorem 2.3: texier2021topologyrealalgebraiccurves, Viro
  • Remark 2.4
  • Definition 2.5: BIMS15
  • Proposition 2.6: BIMS15,texier2021topologyrealalgebraiccurves
  • Theorem 2.7: BIMS15,Viro
  • Definition 2.8: texier2021topologyrealalgebraiccurves
  • ...and 17 more