Remark on the roots of generalized Lens equations

Abstract

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