The membership problem for polynomial ideals in terms of residue currents

Abstract

We find a relation between the vanishing of a globally defined residue current on ${\P }^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther's theorem, and we also obtain a generalization of that theorem. There are also connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.