arXiv subjects

Zoe Geiselmann

Publications and source records attributed to Zoe Geiselmann.

2 recordsLinked to original sources

arXiv2026

Fast Isotopy Computation for T-Curves

A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot \Delta_2$ 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.

math.AG↗

arXiv2026

Limits of combinatorial patchworking

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).

math.AG↗