Sharp bounds for the valence of certain logharmonic polynomials

TL;DR

This paper proves that the maximum valence of certain logharmonic polynomials with a linear factor is exactly 3n-1, confirming a conjecture and completing the understanding of their valence bounds.

Contribution

It establishes the sharpness of the upper bound for the valence of logharmonic polynomials with a linear factor, resolving a conjecture in the field.

Findings

The valence bound of 3n-1 is sharp for these polynomials.

The result confirms the conjecture of Bshouty and Hengartner.

It completes the characterization of valence bounds for this class of polynomials.

Abstract

Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner.