Multiplicity operators

For functions of a single complex variable, zeros of multiplicity greater than k are characterized by the vanishing of the first k derivatives. There are various quantitative generalizations of this statement, showing that for functions that are in some sense close to having a zero of multiplicity greater than k, the first k derivatives must be small. In this paper we aim to generalize this situation to the multi-dimensional setting. We define a class of differential operators, the multiplicity operators, which act on maps from ℂⁿ to ℂⁿ and satisfy properties analogous to those described above. We demonstrate the usefulness of the construction by applying it to some problems in the theory of Noetherian functions.