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arXiv · 2602.06888

Limits of combinatorial patchworking

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

It is shown that there are real plane algebraic curves of degree eight that cannot be realized as T-curves, i.e., via combinatorial patchworking. In fact, this holds for several real schemes (i.e., ambient isotopy types) with the maximal number of real components, called $M$-curves. On the other hand, each nonempty real scheme of lower degree, maximal or not, arises as a T-curve. By constructing one patchwork of the dilated triangle $d\cdot\Delta_2$ for each nonempty real scheme of degree $d\leq 7$, we provide an explicit method for constructing polynomials realizing these real schemes. This resolves a question of Itenberg and Viro (1996).

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Zoe Geiselmann, Michael Joswig, Lars Kastner, Konrad Mundinger, Sebastian Pokutta, Christoph Spiegel, Marcel Wack, Max Zimmer. 2026-02-06. Limits of combinatorial patchworking. https://arxiv.org/abs/2602.06888

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