Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

Let ${\mathcal{H}}_d^{(t)}$ ($t\ge -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball ${\mathbb{B}}_d$ with kernel $$k(z,w)=\frac{1}{(1-⟨z,w⟩{)}^{d+t+1}}\mathrm{.}$$ We prove that if an ideal $I \triangleleft $\mathbb{C}$[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the “approximate stable division property”, then the closure of $I$ in ${\mathcal{H}}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.