# Remark on the roots of generalized Lens equations

## TL;DR

This paper investigates the roots of generalized Lens polynomials and demonstrates that the maximum number of roots can be achieved with a specific form of these polynomials, refining previous results using bifurcation techniques.

## Contribution

It shows that the maximum number of roots for generalized Lens polynomials can be attained with a modified bifurcation family, extending prior work on harmonic splitting Lens polynomials.

## Key findings

- Maximum roots for generalized Lens polynomials can be achieved with a specific construction.
- A slight modification of bifurcation families suffices to reach the maximum number of roots.
- The result extends previous bounds on roots of harmonic Lens type polynomials.

## Abstract

We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$ and with ${\rm deg}\,q(z)=n$, ${\rm deg}\,r(\bar z)=m$ and ${\rm deg}\,p(z)\le n$. We have shown that there exists a harmonically splitting polynomial $r(\bar z)q(z)-p(z)$ which takes $5n+m-6$ roots, using a bifurcation family of polynomials. In this note, we show that this number can be taken by a generalized Lens polynomial ${\bar z}^mq(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomial.