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<p>In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((z-w)^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>-</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.</p>

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On Numerical Invariants for Submodules \([(z-w)^2]\) in \(H^2(\mathbb {D}^2)\)

  • 期刊论文

  • 发表日期：

    2026年6月22日

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  • Yin Liu,
  • Yufeng Lu,
  • Chao Zu

摘要

In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial \((z-w)^2\) ( z - w ) 2 and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.