Fast Isotopy Computation for T-Curves
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913021235822592 |
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| author | Geiselmann, Zoe Joswig, Michael Kastner, Lars Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Wack, Marcel Zimmer, Max |
| author_facet | Geiselmann, Zoe Joswig, Michael Kastner, Lars Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Wack, Marcel Zimmer, Max |
| contents | A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot Δ_2$ together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast Isotopy Computation for T-Curves Geiselmann, Zoe Joswig, Michael Kastner, Lars Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Wack, Marcel Zimmer, Max Algebraic Geometry Computational Geometry Combinatorics 14P25, 14Q05, 14Q30, 52B20 A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot Δ_2$ together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves. |
| title | Fast Isotopy Computation for T-Curves |
| topic | Algebraic Geometry Computational Geometry Combinatorics 14P25, 14Q05, 14Q30, 52B20 |
| url | https://arxiv.org/abs/2604.09221 |