On numerical invariants for submodules $[(z-w)^2]$ in $H^2(\mathbb{D}^2)$

TL;DR

This paper investigates numerical invariants of a specific homogeneous submodule in the Hardy space over the bidisk, providing explicit formulas and analyzing their properties through a detailed example.

Contribution

It offers explicit formulas for invariants of the submodule generated by (z-w)^2 and demonstrates their monotonicity, extending understanding beyond linear cases.

Findings

Derived explicit formulas for the invariants.

Verified the monotonicity property in this setting.

Provided a detailed example illustrating invariant behavior.

Abstract

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 $(z-w)^2$ 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.