Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules [zk−wℓ]
Abstract
Let Mk,ℓ=[zk−wℓ]⊂ H2(𝔻2), k,ℓ∈ℕ, k≠ℓ . We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that Σj(Mk,ℓ) = Ck,ℓ/j + O(j−2), where Ck,ℓ = ∫0∞ x2/((x+1/k)2(x+1/ℓ)2) dx . The first strict descent together with Ck,ℓ recovers the unordered pair {k,ℓ}, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction Σj(Mk,ℓ) = Ck,ℓ/j + Ψk,ℓ(j)/j2 + O(j−3), whose least period is lcm(k,ℓ), and we determine the leading amplitudes of the strict drops. For Yang's generating function 𝒫k,ℓ(t) = ∑j=0∞Σj(Mk,ℓ)tj, we prove that it has radius of convergence one and a logarithmic singularity at t=1; hence it is never a polynomial for k≠ℓ, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair {k,ℓ}, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.
Related
- Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules [zk−wℓ]
- Symbolic Powers and Asymptotic Invariants of GL-Invariant Ideals
- Invariants of big classes from asymptotic total Betti numbers
- Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings
- Grothendieck ring of varieties and at-the-characteristic numerical invariants
BibTeX
@misc{liu2026asymptotic,
title = {{Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules $[z^k-w^\ell]$}},
author = {Yin Liu and Yufeng Lu and Yixin Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36516},
eprint = {2609.36516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36516},
}
RIS
TY - UNPB
TI - Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules [z^k−w^ℓ]
AU - Yin Liu
AU - Yufeng Lu
AU - Yixin Yang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36516
UR - https://arxiv.org/abs/2609.36516
AB - Let M_k,ℓ=[z^k−w^ℓ]⊂ H^2(𝔻^2), k,ℓ∈ℕ, k≠ℓ . We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that Σ_j(M_k,ℓ) = C_k,ℓ/j + O(j^−2), where C_k,ℓ = ∫_0^∞ x^2/((x+1/k)^2(x+1/ℓ)^2) dx . The first strict descent together with C_k,ℓ recovers the unordered pair {k,ℓ}, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction Σ_j(M_k,ℓ) = C_k,ℓ/j + Ψ_k,ℓ(j)/j^2 + O(j^−3), whose least period is lcm(k,ℓ), and we determine the leading amplitudes of the strict drops. For Yang's generating function 𝒫_k,ℓ(t) = ∑_j=0^∞Σ_j(M_k,ℓ)t^j, we prove that it has radius of convergence one and a logarithmic singularity at t=1; hence it is never a polynomial for k≠ℓ, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair {k,ℓ}, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.
ER -