# The maximum number of zeros of $r(z) - \overline{z}$ revisited

## TL;DR

This paper establishes bounds on the maximum number of zeros for rational harmonic functions of a specific form, depending on polynomial degrees, and proves the regularity of functions attaining these bounds.

## Contribution

It generalizes previous results by deriving degree-dependent bounds and proving regularity for extremal functions.

## Key findings

- Derived bounds on zeros depending on polynomial degrees
- Proved regularity of functions reaching these bounds
- Extended previous literature on rational harmonic functions

## Abstract

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions $f(z) = \frac{p(z)}{q(z)} - \overline{z}$, which depend on both $\mathrm{deg}(p)$ and $\mathrm{deg}(q)$. Furthermore, we prove that any function that attains one of these upper bounds is regular.