Multiplicity Operators

TL;DR

This paper introduces multiplicity operators for multivariate functions, generalizing the concept of derivative-based multiplicity characterization from single-variable to multi-dimensional settings, with applications to Noetherian functions.

Contribution

It defines a new class of differential operators, multiplicity operators, that extend multiplicity concepts to multivariate functions and demonstrates their usefulness in complex analysis.

Findings

Multiplicity operators characterize higher multiplicity points in multivariate functions.

Application to Noetherian functions shows practical utility of the operators.

Provides a framework for quantitative analysis of multiplicity in multiple dimensions.

Abstract

For functions of a single complex variable, points 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 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 \emph{multiplicity operators}, which act on maps from $\C^n$ to $\C^n$ 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.