Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules $[z^k-w^\ell]$

Abstract

We study Yang's higher numerical invariants for the quasi-homogeneous submodules \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\neq\ell. \] 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 $z^k-w^\ell$, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For $j\geq1$, set \[ A_j=\left\lceil\frac{j}{\ell}\right\rceil, \qquad B_j=\left\lceil\frac{j}{k}\right\rceil. \] We obtain \[ Σ_0(M_{k,\ell})=\frac{π^2}{6}, \qquad Σ_j(M_{k,\ell})=F(A_j,B_j), \] where $F$ is an explicitly evaluated symmetric function on $\mathbb N^2$. The sequence is nonincreasing, tends to zero, and satisfies $Σ_j>Σ_{j+1} \Longleftrightarrow k\mid j\ \text{or}\ \ell\mid j$. Hence its strict descent set is $k\mathbb N\cup\ell\mathbb N$. This descent set determines the additive semigroup $\langle k,\ell\rangle$, while the complete numerical invariant sequence determines the unordered pair $\{k,\ell\}$. The order of the two exponents is not recoverable because of coordinate-swap symmetry.