T-Curve

A T-curve is a real algebraic curve obtained by Viro's combinatorial patchworking from a regular unimodular triangulation of a lattice polygon with a sign assigned to every lattice point (Viro 1984). For plane curves of polynomial degree , the polygon is the triangle with vertices , , and . After reflecting the signed triangulation into the coordinate quadrants with the parity rule, midsegments separating opposite signs form the patchworked curve. Regularity of the triangulation ensures an algebraic realization.

A maximal T-curve attains the Harnack's theorems bound on real connected components. Its real scheme records the ambient isotopy type in the real projective plane, including nesting of ovals and the pseudoline present in odd degree. Geiselmann et al. (2026) enumerated exactly 38 real schemes of maximal degree-8 T-curves, each with 22 real components, and 225 maximal degree-9 schemes, each with 29 components. The computation certifies regular triangulation witnesses for the returned schemes. These are classifications within combinatorial patchworking, not classifications of all real plane algebraic curves of those degrees.