Bounds for the degrees in the division problem (Q914896)

The authors study the division problem. Namely, let \(f_ 1,...,f_ m\in {\mathbb{C}}[z]\) \((z=(z_ 1,...,z_ n))\), and let I be the ideal they generate. Then for \(f\in I\), what can be said about the polynomial \(q_ 1,...,q_ m\) that satisfy \(f=q_ 1f_ 1+q_ 2f_ 2+...+q_ mf_ m\). They obtain some results on the degree of \(q_ i\) under a certain assumption. The proof relies on \({\bar \partial}\)-theory.