A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot Δ_2$ 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.

翻译：给定一个度为$d$的T-曲线，它由$d \cdot \Delta_2$的正则单形三角剖分及其格点上的符号分布所定义。根据Viro的补丁定理，这决定了相同次数的光滑实平面投影代数曲线的环境同痕类型（即实方案）。我们提出了一种近二次时间复杂度的算法，用于从三角剖分和符号中提取该同痕类型。通过GPU加速实现，该算法每秒可计算数十亿个实方案，从而支持大规模穷举枚举。该算法对于我们近期通过T-曲线构造所有121个七次实方案的工作至关重要。