Perturbing rational harmonic functions by poles

Abstract

We study how adding certain poles to rational harmonic functions of the form $R(z)-\bar{z}$, with $R(z)$ rational and of degree $d\ge 2$, affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing (ArXiv e-print 2003). Of particular interest is the construction and the behavior of rational functions $R(z)$ that are {\em extremal} in the sense that $R(z)-\bar{z}$ has the maximal possible number of $5(d-1)$ zeros.

Minor corrections, better color scheme for phase portraits

References in corpus (2)

• Mathematics of Gravitational Lensing: Multiple Imaging and Magnification
• Sharp parameter bounds for certain maximal point lenses

Cited by in corpus (11)

• A Note on the Maximum Number of Zeros of $r(z)-\bar{z}$
• Sharp parameter bounds for certain maximal point lenses
• Creating images by adding masses to gravitational point lenses
• A Newton method for harmonic mappings in the plane
• The maximum number of zeros of $r(z)-\overline{z}$ revisited
• How constant shifts affect the zeros of certain rational harmonic functions
• The index of singular zeros of harmonic mappings of anti-analytic degree one
• On the zeros of polyanalytic polynomials
• On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force
• Multiplane gravitational lenses with an abundance of images
• The transport of images method: computing all zeros of harmonic mappings by continuation