Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules [zk−wℓ]
Abstract
We study Yang's higher numerical invariants for the quasi-homogeneous submodules Mk,ℓ=[zk−wℓ]⊂ H2(𝔻2), k,ℓ∈ℕ, k≠ℓ. The Hilbert--Schmidt property and the low-order invariants Σ0 and Σ1 for this family follow from earlier results on Mθ,φ-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with zk−wℓ, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For j≥1, set Aj=⌈j/ℓ⌉, Bj=⌈j/k⌉. We obtain Σ0(Mk,ℓ)=π2/6, Σj(Mk,ℓ)=F(Aj,Bj), where F is an explicitly evaluated symmetric function on ℕ2. The sequence is nonincreasing, tends to zero, and satisfies Σj>Σj+1 Longleftrightarrow k| j or ℓ| j. Hence its strict descent set is kℕ∪ℓℕ. This descent set determines the additive semigroup ⟨ k,ℓ⟩, while the complete numerical invariant sequence determines the unordered pair {k,ℓ}. The order of the two exponents is not recoverable because of coordinate-swap symmetry.
Related
- Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules [zk−wℓ]
- Grothendieck ring of varieties and at-the-characteristic numerical invariants
- Numerical invariants for three-generated ideals
- Direct Vertex-Derivative Coefficient Propagation for Homogeneous Numerical Integration (HNI)
- Integral Invariants of Vasudeva Murthy's Relaxation Systems: Analysis and Numerical Validation
BibTeX
@misc{liu2026numerical,
title = {{Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules $[z^k-w^\ell]$}},
author = {Yin Liu and Yufeng Lu and Yixin Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25640},
eprint = {2609.25640},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25640},
}
RIS
TY - UNPB
TI - Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules [z^k−w^ℓ]
AU - Yin Liu
AU - Yufeng Lu
AU - Yixin Yang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25640
UR - https://arxiv.org/abs/2609.25640
AB - We study Yang's higher numerical invariants for the quasi-homogeneous submodules M_k,ℓ=[z^k−w^ℓ]⊂ H^2(𝔻^2), k,ℓ∈ℕ, k≠ℓ. The Hilbert--Schmidt property and the low-order invariants Σ_0 and Σ_1 for this family follow from earlier results on M_θ,φ-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with z^k−w^ℓ, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For j≥1, set A_j=⌈j/ℓ⌉, B_j=⌈j/k⌉. We obtain Σ_0(M_k,ℓ)=π^2/6, Σ_j(M_k,ℓ)=F(A_j,B_j), where F is an explicitly evaluated symmetric function on ℕ^2. The sequence is nonincreasing, tends to zero, and satisfies Σ_j>Σ_j+1 Longleftrightarrow k| j or ℓ| j. Hence its strict descent set is kℕ∪ℓℕ. This descent set determines the additive semigroup ⟨ k,ℓ⟩, while the complete numerical invariant sequence determines the unordered pair {k,ℓ}. The order of the two exponents is not recoverable because of coordinate-swap symmetry.
ER -