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).
From: Marcel Wack [view email]
[v1]
Fri, 6 Feb 2026 17:21:11 UTC (201 KB)
[v2]
Mon, 23 Feb 2026 12:52:22 UTC (69 KB)
[v3]
Mon, 27 Jul 2026 09:46:13 UTC (936 KB)
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