Remark on the roots of generalized Lens equations

We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$andalsoharmonicallysplittingLenstypepolynomial$L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$andwith${\rm deg},q(z)=n$,${\rm deg},r(\bar z)=m$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 ${\bar{z}}^{m}q(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomial.