2023/05/15 by Alemu, Oluma Ararso
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.08606
The valence of a function f at a point z0 is the number of distinct, finite solutions to f(z) = z0. In this paper, we bound the valence of complex-valued harmonic polynomials in the plane for some special harmonic polynomials of the form f(z) =p(z)q(z), where p(z) is an analytic polynomial of degree n and q(z) is an analytic polynomial of degree m, and q(z) ≠ αp(z) for some constant α. Using techniques of complex dynamics used in the work Sheil-Small and Wilmshurst on the valence of harmonic polynomials, we prove that the harmonic polynomial f(z) = p(z)q(z) has the valency of m + n.