On the roots of an extended lens equation and an application

TL;DR

This paper studies roots of a generalized extended lens equation, constructs specific polynomial deformations, and applies findings to analyze the topology of moduli spaces of certain weighted homogeneous polynomials with isolated singularities.

Contribution

It introduces a new class of mixed polynomials with controlled roots and extends Rhie's example, applying these results to the topology of moduli spaces of weighted homogeneous polynomials.

Findings

Constructed a mixed polynomial with 5n zeros

Generalized Rhie's example through polynomial deformation

Applied results to moduli space component analysis

Abstract

We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p<n$ whose numerator is a mixed polynomial of degree $(n+m;n,m)$. Then we consider its deformation of type $L_{n,m}+\epsilon/z^m$ to construct a special mixed polynomial of degree $(n+2m;n+m,m)$ with $5n$ zeros. This generalizes an example of Rhie. We give an application to the number of connected components of the moduli space of strongly mixed weighted homogeneous polynomials of two variables with isolated singularity at the origin.