# The index of singular zeros of harmonic mappings of anti-analytic degree one

## TL;DR

This paper investigates the index of singular zeros in harmonic mappings of the form f(z)=h(z)-conjugate(z), providing a characterization that allows direct computation from the power series of h, thus extending understanding of zeros on the critical set.

## Contribution

It introduces a novel characterization of the index of singular zeros in harmonic mappings, enabling direct calculation from the power series of h.

## Key findings

- Characterization of the index of singular zeros.
- Method to determine the index from power series of h.
- Extension of zero index properties to critical set zeros.

## Abstract

We study harmonic mappings of the form $f(z) = h(z) - \overline{z}$, where $h$ is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of $f$, where the Jacobian of $f$ is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros. Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of $h$.