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

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel $$k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} .$$ 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 $p>2$.