On the number of zeros of certain rational harmonic functions. (English) Zbl 1090.30008
Let \(r\) be a rational function of degree \(n>1.\) It is shown that the function \(\overline{r(z)}-z\) has at most \(5n-5\) zeros (in the complex plane). D. Khavinson and G. Swiatek [Proc. Am. Math. Soc. 131, 409-414 (2003; Zbl 1034.30003)] had previously shown that the number of zeros is at most \(3n-2\) if \(r\) is a polynomial. The proofs in both papers use, among other things, a result from complex dynamics. The result has a very interesting application to gravitational lensing, giving an upper bound for the maximum number of lensed images due to an \(n\)-point gravitational lens. This bound had been conjectured by S. H. Rhie.
Reviewer: Walter Bergweiler (Kiel)
MSC:
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 83C99 | General relativity |
Keywords:
harmonic function; rational function; Wilmshurst conjecture; gravitational lensing; S. H. Rhie conjectureCitations:
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