Fast Isotopy Computation for T-Curves
career value
191K/year
🤖 AI Summary
This study addresses the problem of efficiently computing the ambient isotopy types—i.e., real configurations—of real algebraic curves defined by T-curves. Building upon Viro’s patchworking theorem, the authors propose an algorithm with near-quadratic time complexity that starts from a given regular unimodular triangulation and its associated sign distribution on lattice points. By integrating GPU-based parallel acceleration, the method achieves unprecedented computational throughput, enabling the enumeration of billions of real configurations per second. As a major application, this approach yields the first complete enumeration of all 121 real configurations for septic (degree-7) T-curves, substantially advancing the feasibility and efficiency of exhaustive configuration searches for high-degree real algebraic curves.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
T-curves
isotopy type
real scheme
Viro's Patchworking
plane algebraic curves
Innovation
Methods, ideas, or system contributions that make the work stand out.
T-curves
isotopy computation
Viro's patchworking
GPU acceleration
real algebraic curves
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