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

Let H _d ^(t) (t ≥ −d, t > −3) be the reproducing kernel Hilbert space on the unit ball B _d with kernel 1 k(z, w) = _{(1 − h} z, wi _)d +t+ ₁ . We prove that if an ideal I C[z ₁ , . . ., z _d ] (not necessarily homogeneous) has what we call the approximate stable division property, then the closure of I in 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 C[x, y] is p-essentially normal for p > 2.