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.
🧑🎓 Project Members
spiegel (at) zib.de
mundinger (at) zib.de
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.