On the valence of logharmonic polynomials

Abstract

Investigating a problem posed by W. Hengartner (2000), we study the maximal valence (number of preimages of a prescribed point in the complex plane) of logharmonic polynomials, i.e., complex functions that take the form $f(z)=p(z)\overline{q(z)}$ of a product of an analytic polynomial $p(z)$ of degree $n$ and the complex conjugate of another analytic polynomial $q(z)$ of degree $m$. In the case $m=1$, we adapt an indirect technique utilizing anti-holomorphic dynamics to show that the valence is at most $3n-1$. This confirms a conjecture of Bshouty and Hengartner (2000). Using a purely algebraic method based on Sylvester resultants, we also prove a general upper bound for the valence showing that for each $n,m\ge 1$ the valence is at most $n^2+m^2$. This improves, for every choice of $n,m\ge 1$, the previously established upper bound $(n+m{)}^2$ based on Bezout's theorem. We also consider the more general setting of polyanalytic polynomials where we show that this latter result can be extended under a nondegeneracy assumption.

15 pages, 2 figures. Updated version with minor revisions. This version also includes visualization of Julia sets (Figure 1) kindly provided by Walter Bergweiler. The paper appeared in the volume "Recent Progress in Function Theory and Operator Theory" of the book collection in Contemporary Mathematics