On the valence of logharmonic polynomials

TL;DR

This paper investigates the maximum number of preimages (valence) of logharmonic polynomials, confirming a conjecture for a special case and establishing new upper bounds using algebraic and dynamical methods.

Contribution

It proves a conjecture on valence for logharmonic polynomials with m=1 and derives a general algebraic upper bound for valence for all degrees n,m.

Findings

Valence is at most 3n-1 for m=1, confirming a conjecture.

General upper bound for valence is n^2 + m^2, improving previous bounds.

Extension of results to polyanalytic polynomials under nondegeneracy conditions.

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 $n,m \geq 1$ the valence is at most $n^2+m^2$. This improves, for every choice of $n,m \geq 1$, 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.