Fast Isotopy Computation for T-Curves

TL;DR

The paper introduces a fast GPU-accelerated algorithm for computing the isotopy type of T-curves from triangulations and signs, enabling large-scale enumeration of real algebraic curves.

Contribution

It presents a near-quadratic time algorithm for isotopy computation and demonstrates its scalability and application in enumerating real schemes of degree seven.

Findings

Algorithm achieves near-quadratic time complexity.

GPU implementation enables billions of computations per second.

Facilitated the enumeration of all 121 real schemes of degree seven.

Abstract

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.