Fast Isotopy Computation for T-Curves

Abstract: 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.

• The paper introduces a near-quadratic algorithm that extracts the real scheme of a T-curve, bypassing traditional expensive symbolic methods.
• It leverages combinatorial patchworking and GPU-parallelizable union-find techniques to classify curves at rates up to 10^8 patchworks per second.
• Results reveal complete realizability for degree ≤7 T-curves and notable limitations for degree 8, offering new insights into Hilbert’s 16th problem.

Fast Isotopy Computation of T-Curves: Technical Summary and Implications

Overview

The paper "Fast Isotopy Computation for T-Curves" (2604.09221) introduces a near-quadratic (in degree $d$) time algorithm for extracting the ambient isotopy type (real scheme) of a smooth real plane projective algebraic curve given as a T-curve, specified by a regular unimodular triangulation of $d\cdot\Delta_2$ with an associated sign distribution. Leveraging combinatorial patchworking and an efficient GPU-parallelizable implementation, the method enables exhaustive enumeration of T-curve real schemes at unprecedented scale and speed, with concrete applications to classical questions in real enumerative geometry, notably Hilbert’s 16th problem.

Mathematical and Algorithmic Foundations

A T-curve is constructed by Viro's patchworking from a regular unimodular triangulation $T$ of the $d$-fold dilated simplex $\Delta_2$ and a sign assignment $\sigma$ on the lattice points of $d\cdot\Delta_2$. The patchworking encodes the topology of the real algebraic curve with only combinatorial data. The main computational challenge addressed by the paper is the extraction of the real scheme---the nested configuration of ovals and pseudoline---from this data, bypassing the prohibitively expensive symbolic methods such as cylindrical algebraic decomposition.

The algorithm innovatively operates on the reflected, signed triangulation $T^\diamond$, efficiently identifying:

• Monochromatic connected components: Computed via union-find on non-crossed edges (edges with equal endpoint signs).
• Region identification: By merging antipodal boundary vertices and edges, corresponding to global topology modding out projective identifications.
• Adjacency and root detection: The region graph is built via crossed edges (those separating opposite signs), forming a tree reflecting the structure of the real scheme. For even $d$, the root is the unique non-orientable region forming an odd cycle under antipodal identification; for odd $d$, it is the region with a self-loop reflecting the unique pseudoline.

The algorithm achieves a complexity of $d\cdot\Delta_2$0 per patchwork, where $d\cdot\Delta_2$1 is the inverse Ackermann function, stemming from the union-find operations. This marks a substantial improvement compared to prior naïve cubic or even quintic approaches.

Implementation and Exhaustive Enumeration

Two open-source implementations are offered: a minimal C++ library (libisotopy) and a GPU-accelerated Rust toolkit supporting CUDA and Metal, designed for large-scale enumeration. The design exploits the fixed, degree-bounded memory footprint and fully parallel structure of the problem, allowing classification rates of $d\cdot\Delta_2$2 patchworks per second per GPU for degree 8.

Exhaustive enumeration experiments yield the following strong and, to some extent, contradictory findings:

• For degrees $d\cdot\Delta_2$3, all nonempty real schemes are realized by T-curves, often from a single or few triangulations.
• For degree 8, only $d\cdot\Delta_2$4 of the $d\cdot\Delta_2$5 known maximal oval real schemes are realized as T-curves; many maximum types (including some with known algebraic realizations) are unattainable even with extensive searches. Furthermore, $d\cdot\Delta_2$6 distinct real schemes were constructed, spanning all oval counts, yet this is conjectured to be substantially below the true number for degree 8. This points to an expressive limitation in combinatorial patchworking in higher degrees.

Theoretical and Practical Implications

The algorithm substantially advances the computational tractability of classifying real algebraic curves up to isotopy via combinatorial data. This unlocks novel avenues for studying the realizability landscape of real schemes, enabling:

• Systematic enumeration of real schemes for given degrees and insight into the distribution of oval numbers, nesting, and symmetries.
• Resolution of degree-7 cases for Hilbert’s 16th problem, with all $d\cdot\Delta_2$7 nonempty real schemes realized by T-curves, confirming the completeness of combinatorial patchworking up to this degree.
• Identification of limitations of T-curves for $d\cdot\Delta_2$8, revealing a qualitative change in the patchworking landscape.

Practically, the extreme throughput achieved by the implementations permits large-scale statistical experiments and symmetry reduction analysis, and invites further exploration of degenerate and sparse classes in the search space.

Prospects and Future Directions

Several directions are highlighted for further research:

• Complex schemes: While the current approach focuses on real schemes, extracting finer invariants (dividing type, complex orientations) remains an open computational challenge, with only partial algorithmic work available in the literature.
• Beyond T-curves: The expressive limitations observed for $d\cdot\Delta_2$9 suggest a need for generalizations or alternative constructive frameworks (e.g., more general tropical patchworking, non-patchworked constructions, or higher-dimensional analogues).
• Exhaustivity and coverage: Further large-scale searches, possibly with smarter sampling or combinatorial heuristics, may close the gap between realized and possible schemes for higher degrees.
• Integration with symbolic/numerical homotopy techniques: Combining combinatorial and effective field theories could enhance both the constructive and classification capabilities for real algebraic topology.

Conclusion

The presented algorithm forms a new benchmark in computational real algebraic geometry, combining combinatorial theory, algorithmic ingenuity, and parallel implementation to clarify the landscape of realizable real schemes via T-curves. Its impact is twofold: as a tool for exhaustive classification up to medium degree, and as a diagnostic for the expressive power and limitations of combinatorial patchworking---informing both the theoretical study of Hilbert’s 16th problem and the development of future constructive and computational techniques in the field.