Number and location of pre-images under harmonic mappings in the plane

Författare

  • Olivier Sète TU Berlin, Department of Mathematics, MA 3-3
  • Jan Zur TU Berlin, Department of Mathematics, MA 3-3

Nyckelord:

harmonic mappings, pre-images, caustics, argument principle, valence, zeros of harmonic polynomials

Abstract

 

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping \(f\), using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under \(f\) geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every \(k = n, n+1, \ldots, n^2\) there exists a harmonic polynomial of degree \(n\) with \(k\) zeros.
Sektion
Articles

Publicerad

2021-06-21

Referera så här

Sète, O., & Zur, J. (2021). Number and location of pre-images under harmonic mappings in the plane. Annales Fennici Mathematici, 46(1), 225–247. Hämtad från https://afm.journal.fi/article/view/109574