双圆盘子模 $[z^k-w^\ell]$ 的 Yang 数值不变量的渐近刚性与边界结构
Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules $[z^k-w^\ell]$
- Nanyang Normal University（南阳师范学院）
- Dalian University of Technology（大连理工大学）
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AI总结：
本文研究双圆盘子模的 Yang 数值不变量的渐近行为，证明其主项系数恢复指标对，并揭示生成函数具有对数奇异性，否定多项式性猜想。
AI中文摘要：
设 $M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2)$，其中 $k,\ell\in\mathbb N$ 且 $k\ne\ell$。我们研究 Yang 数值不变量的指标渐近行为及其生成函数的边界结构。从我们先前工作中获得的精确阶梯公式出发，我们证明 $\Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + O(j^{-2})$，其中 $C_{k,\ell} = \int_0^\infty \frac{x^2}{(x+1/k)^2(x+1/\ell)^2}\\,dx$。首次严格下降与 $C_{k,\ell}$ 共同恢复无序对 $\{k,\ell\}$，从而得到渐近刚性原理。在下一阶，我们获得周期修正 $\Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + \frac{\Psi_{k,\ell}(j)}{j^2} + O(j^{-3})$，其最小周期为 $\operatorname{lcm}(k,\ell)$，并确定严格下降的前导振幅。对于 Yang 生成函数 $\mathcal P_{k,\ell}(t) = \sum_{j=0}^{\infty}\Sigma_j(M_{k,\ell})t^j$，我们证明其收敛半径为 1，且在 $t=1$ 处具有对数奇异性；因此当 $k\ne\ell$ 时它绝不是多项式，这在此拟齐次族内对 Yang 的多项式性问题给出了否定回答。周期性的高阶修正产生单位根多对数边界模式。前导对数系数与二阶边界支撑共同确定无序对 $\{k,\ell\}$，而逐次重整化的边界极限恢复每个有限阶周期渐近项的傅里叶系数。
英文摘要：
Let \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\ne\ell . \] 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 \[ Σ_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + O(j^{-2}), \] where \[ C_{k,\ell} = \int_0^\infty \frac{x^2} {(x+1/k)^2(x+1/\ell)^2}\,dx . \] The first strict descent together with $C_{k,\ell}$ recovers the unordered pair $\{k,\ell\}$, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction \[ Σ_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + \frac{Ψ_{k,\ell}(j)}{j^2} + O(j^{-3}), \] whose least period is \(\operatorname{lcm}(k,\ell)\), and we determine the leading amplitudes of the strict drops. For Yang's generating function \[ \mathcal P_{k,\ell}(t) = \sum_{j=0}^{\infty}Σ_j(M_{k,\ell})t^j, \] we prove that it has radius of convergence one and a logarithmic singularity at $t=1$; hence it is never a polynomial for $k\ne\ell$, 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 $\{k,\ell\}$, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.
