Algorithm for Computing the Motivic Hilbert Zeta Function of Monomial Curve Singularities

Core Concepts

This research paper presents an algorithm for computing the motivic Hilbert zeta function of curve singularities, particularly those with a monomial local ring, by leveraging the structure of the punctual Hilbert scheme and its stratification.

Abstract

• Bibliographic Information: Zhu, W., Chen, Y., & Mourtada, H. (2024). Algorithm for motivic Hilbert zeta function of monomial curves. arXiv preprint arXiv:2411.03283v1.

• Research Objective: To develop an efficient algorithm for computing the motivic Hilbert zeta function for curve singularities with a monomial local ring.

• Methodology: The researchers utilize the properties of the punctual Hilbert scheme, a parameter space for ideals in the local ring of a curve singularity. They exploit the stratification of this scheme, where each stratum corresponds to ideals with a specific valuation set. By analyzing the relationships between these strata, represented as a tree structure derived from the valuation semigroup, the algorithm efficiently computes the motivic class of each stratum and ultimately the motivic Hilbert zeta function.

• Key Findings:

  • The paper establishes a connection between the edges of the tree structure, representing the relationship between strata of the punctual Hilbert scheme, and a geometric morphism between these strata.
  • This morphism allows for the computation of the motivic class of a stratum based on the class of its parent stratum in the tree, leading to an iterative algorithm.
  • The authors provide a theoretical bound for truncating the infinite valuation semigroup to ensure the algorithm's computational feasibility.

• Main Conclusions: The proposed algorithm effectively computes the motivic Hilbert zeta function for monomial curve singularities. The authors provide a Python implementation of the algorithm, demonstrating its practicality.

• Significance: This research contributes significantly to computational algebraic geometry by providing an efficient method for calculating a complex invariant of curve singularities. The motivic Hilbert zeta function has connections to knot invariants and other geometric and topological properties, making this algorithm potentially valuable for various applications.

• Limitations and Future Research: The algorithm currently focuses on monomial curve singularities. Future research could explore extending this approach to broader classes of curve singularities or investigating further connections between the motivic Hilbert zeta function and other invariants in singularity theory.

Stats

The conductor 'c' of the valuation semigroup is a crucial parameter in the algorithm.
The algorithm utilizes a truncation bound 'n', set to (c-1)(α1+2), to handle the infinite valuation semigroup computationally.
The example demonstrates the algorithm's application to a curve singularity with a local ring C[[t^4, t^5, t^6]], where the conductor 'c' is 8.

Quotes

"The goal of this paper is to develop an algorithm to compute the motivic Hilbert zeta function ZHilb(C,O)(q) ∈K0(V arC)[[t]] for a germ of singular curve (C, O) with monomial complete local ring."
"The main computational challenge arises from the infinity of Γ. To address this, we approximate Γ by truncating it to a finite subset to allow effective algorithm operation."
"The Python implementation of our algorithm is available at https://github.com/whaozhu/motivic_hilbert."

Key Insights Distilled From

Algorithm for motivic Hilbert zeta function of monomial curves

by Wenhao Zhu, ... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03283.pdf

Deeper Inquiries

How might this algorithm be adapted or extended to compute other invariants of curve singularities beyond the motivic Hilbert zeta function?

This algorithm, centered on the computation of the motivic Hilbert zeta function for monomial curve singularities, can be potentially adapted to compute other invariants by leveraging the information encoded in the algorithm's process and output. Here are a few possibilities:

Poincaré Series of the Local Ring: The algorithm meticulously decomposes the punctual Hilbert scheme into strata (C[Δ]) and computes their motivic classes. These classes, in turn, can be used to determine the Poincaré series of the local ring of the singularity. This connection arises from the relationship between the Hilbert function and the Poincaré series.

Alexander Invariants: Given the established link between the motivic Hilbert zeta function and knot invariants for planar curves, this algorithm could potentially be extended to compute Alexander invariants of the knot associated with the curve singularity. This would involve a deeper exploration of the relationship between the stratification of the Hilbert scheme and the topology of the knot.

Hodge-Deligne Polynomials: The algorithm's computation relies heavily on the Grothendieck ring of varieties. This ring is closely related to Hodge-Deligne polynomials, which encode richer information about the cohomology of varieties. Adapting the algorithm to compute these polynomials could provide finer invariants of the singularity.

Motivic Milnor Fiber: The motivic Milnor fiber is a sophisticated invariant of singularities. While this algorithm focuses on the Hilbert scheme, exploring potential connections between the Hilbert scheme stratification and the Milnor fiber could open avenues for computing motivic Milnor fiber invariants.

These are just a few potential directions. The key takeaway is that the algorithm's detailed analysis of the punctual Hilbert scheme, particularly its stratification and the computation of motivic classes, provides a rich foundation for exploring and computing other invariants of curve singularities.

Could there be alternative approaches, perhaps using different stratifications or computational techniques, to compute the motivic Hilbert zeta function for a broader class of singularities?

Yes, alternative approaches to compute the motivic Hilbert zeta function for a broader class of singularities are certainly possible. The current algorithm leverages the specific structure of monomial curve singularities and their associated valuation semigroups. Exploring different stratifications or computational techniques could extend the reach to more general singularities:

Different Stratifications:

Newton Polyhedra: For singularities beyond the monomial case, Newton polyhedra could provide a natural stratification of the Hilbert scheme. The geometry of the Newton polyhedron is closely tied to the singularity's resolution, potentially leading to a combinatorial approach for computing the motivic Hilbert zeta function.
Toric Geometry: If the singularity admits a toric resolution, techniques from toric geometry could be employed. This approach would involve stratifying the Hilbert scheme based on the torus action and utilizing tools from toric geometry to compute the motivic classes of the strata.

Computational Techniques:

Resolution of Singularities: For general singularities, computing the motivic Hilbert zeta function directly might be challenging. However, using a resolution of singularities, one could potentially relate the motivic Hilbert zeta function of the singular variety to that of the resolved variety, which is smooth. This could lead to a recursive approach for computation.
Computer Algebra Systems: Developing more sophisticated algorithms within powerful computer algebra systems could enable the computation of the motivic Hilbert zeta function for more complex singularities. These systems could handle the intricate combinatorial and algebraic manipulations required for such computations.

Beyond Curve Singularities: Extending the computation to higher-dimensional singularities would require significant advancements. Techniques from arc spaces and motivic integration could potentially play a role in this endeavor.

The search for alternative approaches is an active area of research. The key lies in identifying structures or techniques that effectively capture the geometry of the singularity and its Hilbert scheme, ultimately enabling the computation of the motivic Hilbert zeta function.

What are the potential implications of this algorithm for related fields like knot theory or topological data analysis, considering the connections between the motivic Hilbert zeta function and knot invariants?

The algorithm's ability to compute the motivic Hilbert zeta function for monomial curve singularities has intriguing implications for related fields, particularly knot theory and topological data analysis, due to the established connections between the motivic Hilbert zeta function and knot invariants:

Knot Theory:

New Computational Tools: The algorithm provides a new computational tool for exploring knot invariants associated with singular plane curves. By computing the motivic Hilbert zeta function, one gains access to information about the Alexander polynomial and potentially other knot invariants, offering new insights into the topology of knots.
Categorification of Knot Invariants: The algorithm's use of the Grothendieck ring of varieties hints at a possible categorification of knot invariants. This means lifting knot invariants from numerical values to richer algebraic objects, potentially revealing deeper structures and relationships within knot theory.

Topological Data Analysis:

Persistent Homology: The stratification of the Hilbert scheme used in the algorithm could be viewed through the lens of persistent homology. Each stratum represents a different "scale" at which to view the singularity, and the algorithm's computations could provide insights into the persistent homology of the singularity.
Data Analysis of Singularities: The algorithm could contribute to the development of new tools for the topological data analysis of datasets exhibiting singularities. By adapting the algorithm to handle point cloud data, one could potentially extract meaningful information about the topology of singularities present in the data.

Further Connections:

Mirror Symmetry: The motivic Hilbert zeta function has connections to mirror symmetry. This algorithm's computations could potentially shed light on mirror symmetry phenomena associated with curve singularities.
Representation Theory: The appearance of syzygies in the algorithm hints at potential connections to representation theory. Further exploration could reveal relationships between the representation theory of the singularity and its motivic Hilbert zeta function.

The algorithm's impact extends beyond the realm of algebraic geometry. Its ability to compute the motivic Hilbert zeta function, a bridge between algebraic geometry and topology, opens up exciting avenues for research and discovery in related fields like knot theory and topological data analysis.