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 $p>2$.