Papers › Algorithm for motivic Hilbert zeta function of some curve singularities
Algorithm for motivic Hilbert zeta function of some curve singularities
Yizi Chen, Hussein Mourtada, Wenhao Zhu
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by yᵏ=xⁿ with gcd(k,n)=1, and a finer version introduced by the first and third authors together with Mounir Hajli, which currently applies to the specific family yᵏ=xⁿ where gcd(k,n)=1. It is well known that the Hilbert scheme of points on a smooth curve is isomorphic to the symmetric product of the curve. However, the geometry of the Hilbert scheme of points on singular curves remains much less understood. Our algorithms compute the motivic Hilbert zeta functions Z_((C,O))ᴴⁱˡᵇ(q) ∈K₀(Var_ℂ)[[q]], Zm_((C,O))ᴴⁱˡᵇ(a²,q²) ∈K₀(Var_ℂ)[[a², q²]], for such curve singularities, expressed as formal power series with coefficients in the Grothendieck ring of complex varieties. The main computational difficulty arises from the fact that Γ is infinite. To overcome this, we approximate Γ by truncating it to a suitable finite subset, which allows the algorithms to run effectively. We analyze the time complexity of the method and provide an estimate for the effective finite length of Γ required to obtain reliable results. A Python implementation of the algorithms is available at https://github.com/whaozhu/motivic_hilbert.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections