Neural Generative Models for Algebraic Curves ongoing

This project aims to apply neural generative models to real plane algebraic curves to advance the classification of M-curves, a problem from Hilbert's 16th problem that remains open for degree ≥8.

MATH+ EF-LiOpt-1
Sep 2025 to Aug 2028
5
1

🧑‍🎓 Project Members

Michael Joswig
Principal Investigator
joswig (at) math.tu-berlin.de
Sebastian Pokutta
Principal Investigator
pokutta (at) zib.de
Christoph Spiegel
spiegel (at) zib.de
Konrad Mundinger
mundinger (at) zib.de
Max Zimmer
zimmer (at) zib.de

🪙 Funding

This project is being funded by the Berlin Mathematics Research Center MATH+ (project ID EF-LiOpt-1), itself funded by the German Research Foundation (DFG) under Germany's Excellence Strategy (EXC-2046/1, project ID 390685689) from September 2025 to August 2028.

🔬 Project Description

Harnack (1876) showed that a real plane curve of degree d has at most ½(d²−3d+4) ovals; curves achieving this bound are M-curves. Classifying M-curves by isotopy type—the first part of Hilbert’s 16th problem—remains open for d≥8.

This project aims to develop neural generative models for the triangulations and sign distributions used in Viro’s combinatorial patchworking, a technique that encodes curve isotopy types and underlies tropical geometry. Complete enumeration is infeasible and random sampling ineffective at relevant scales, making learned generative approaches essential.

📝 Publications and preprints

Preprints

  1. Geiselmann, Z., Joswig, M., Kastner, L., Mundinger, K., Pokutta, S., Spiegel, C., Wack, M., and Zimmer, M. (2026). 121 Patchworked Curves of Degree Seven. [arXiv]
    [BibTeX]
    @misc{2026_ZoeEtAl_Patchworkedcurves_2602-06888,
  archiveprefix = {arXiv},
  eprint = {2602.06888},
  arxiv = {arXiv:2602.06888},
  primaryclass = {math.AG},
  year = {2026},
  author = {Geiselmann, Zoe and Joswig, Michael and Kastner, Lars and Mundinger, Konrad and Pokutta, Sebastian and Spiegel, Christoph and Wack, Marcel and Zimmer, Max},
  title = {121 Patchworked Curves of Degree Seven},
  date = {2026-02-06}
}