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121 Patchworked Curves of Degree Seven

Zoe Geiselmann, Michael Joswig, Lars Kastner, Konrad Mundinger, Sebastian Pokutta, Christoph Spiegel, Marcel Wack, Max Zimmer

TL;DR

This work settles the degree-$7$ case of Hilbert's problem by proving that every real scheme of degree seven can be realized as a T-curve via combinatorial patchworking on $7\cdot\Delta_2$, using four regular unimodular triangulations. It develops a detailed analysis of sign-distribution equivalence under unimodular symmetries, and constructs explicit infinite families of maximal T-curves (onion, nested box, arrowheads) to realize a broad class of schemes and nesting structures. The degree-six case is detailed as a precursor, showing that two triangulations suffice to realize all 55 nonempty degree-six schemes, with lifting functions and software support provided; degree-seven results extend these methods to achieve full coverage of 121 schemes. The findings demonstrate the power of patchworking to realize all real schemes of degree seven, highlight the central role of the honeycomb triangulation, and pose natural questions about degree eight and the minimal triangulation count required for higher degrees, with implications for both the combinatorial and tropical perspectives in real algebraic geometry.

Abstract

The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle $7\cdotΔ_2$ for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996).

121 Patchworked Curves of Degree Seven

TL;DR

This work settles the degree- case of Hilbert's problem by proving that every real scheme of degree seven can be realized as a T-curve via combinatorial patchworking on , using four regular unimodular triangulations. It develops a detailed analysis of sign-distribution equivalence under unimodular symmetries, and constructs explicit infinite families of maximal T-curves (onion, nested box, arrowheads) to realize a broad class of schemes and nesting structures. The degree-six case is detailed as a precursor, showing that two triangulations suffice to realize all 55 nonempty degree-six schemes, with lifting functions and software support provided; degree-seven results extend these methods to achieve full coverage of 121 schemes. The findings demonstrate the power of patchworking to realize all real schemes of degree seven, highlight the central role of the honeycomb triangulation, and pose natural questions about degree eight and the minimal triangulation count required for higher degrees, with implications for both the combinatorial and tropical perspectives in real algebraic geometry.

Abstract

The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996).
Paper Structure (19 sections, 18 theorems, 25 equations, 10 figures)

This paper contains 19 sections, 18 theorems, 25 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Viro's patchworking
  3. Real schemes and related classifications
  4. Combinatorial patchworking
  5. Combinatorial topology
  6. Regions and nesting
  7. Equivalence of sign distributions
  8. Families of maximal T-curves
  9. Onion curves
  10. Harnack signs and bridge flips
  11. Nested box curves
  12. The arrowheads family
  13. Curves of degree at most six
  14. Curves of degree seven
  15. Concluding remarks
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathcal{T}$ be a regular and unimodular triangulation of $A=d\cdot\Delta_2\cap\mathbb{Z}^2$ with lifting function $\omega:A\to\mathbb{Z}$, and let $\sigma:A\to\mathbb{F}_{2}$ be a sign distribution. Then there is a real number $t_0>0$ such that for all $t\in(0,t_0]$ the real projective plane c is ambient isotopic to the patchworked curve $\mathcal{C}(\mathcal{T},\sigma)$.

Figures (10)

  • Figure 1: Eight T-curves of degree two arising from one unimodular triangulation of $2\cdot\Delta_2$. In each case the real scheme is $\langle 1 \rangle$, and the interior of the unique oval is drawn shaded.
  • Figure 2: Two families of maximal T-curves.
  • Figure 3: A purely combinatorial picture of the propagated quadrangles to the different quadrants. In each subfigure, the top shows the configuration before the flip and the bottom after.
  • Figure 4: Vertex signs at distance at most two from $v$.
  • Figure 5: A bridge flip which affects the root isotopy but not the ambient isotopy.
  • ...and 5 more figures

Theorems & Definitions (35)

  • Theorem 1: Viro's Combinatorial Patchworking Theorem
  • Remark 2
  • Proposition 3
  • proof
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Remark 9
  • ...and 25 more