On the valence of logharmonic polynomials

TL;DR

The paper examines Hengartner's valence problem for logharmonic polynomials $f(z)=p(z)\overline{q(z)}$ with $\deg p=n$, $\deg q=m$ and $p\nparallel q$, delivering two central advances. First, for the case $m=1$, it uses an anti-holomorphic dynamics framework and a generalized argument principle to prove a sharp upper bound of $3n-1$ on the valence, confirming a conjecture of Bshouty and Hengartner. Second, it provides a general, algebraic upper bound of $n^2+m^2$ on the valence for all $n,m \ge 1$ via Sylvester resultants, improving the Bezout bound $(n+m)^2$, and extends to polyanalytic polynomials under a nondegeneracy condition by establishing coprimality of the associated polynomials. The work integrates dynamics, harmonic mappings, and algebraic elimination to tighten universal bounds on finite valence, with numerical results supporting sharpness in low-degree cases and potential relevance to gravitational lensing models. Overall, the paper advances the understanding of extremal valence in polyanalytic settings and provides tools that could influence related problems in complex analysis and applied lensing theory.

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.

Figures (2)

• Figure 1: Left: The basins of attraction for each of the three attracting fixed points of the anti-rational map $z \mapsto \frac{100}{\overline{z}\left[\overline{z}^{2}+3\sqrt{50}\right]}+0.68$ which is an extremal example for the case $n=3$. Right: The basins of attraction for each of the four attracting fixed points of the anti-rational map $z \mapsto \frac{250}{\overline{z}\left[\overline{z}^{3}+4\left(\frac{250}{3}\right)^{3/5}\right]}+0.86$ which is an extremal example for the case $n=4$. In both pictures, regions with the same color/shade are in a common basin of attraction, i.e., are attracted to a common fixed point. Image Credit: Both pictures were created by Walter Bergweiler.
• Figure 2: Left: A depiction of the condition $n^2 + m^2 < N^2$. The monomials of a polyanalytic polynomial $P(z,\overline{z})$ of total degree $N$ correspond to integer lattice points that lie on and within the right triangle. Note that the lattice point $(n,m)$, which is constructed from the maximum degrees in $z$ and $\overline{z}$, need not correspond to an individual monomial. Right: A depiction of a simple sufficient condition $n,m < N/\sqrt{2}$. If the monomials are limited to the pentagonal region, the Resultant Bound is less than the Bezout bound.

Theorems & Definitions (27)

• Conjecture 1.2: Bshouty, Hengartner, 2000
• Theorem 1.3
• Conjecture 1.4: Abdulhadi, Hengartner, 2001
• Theorem 1.5
• Lemma 1.6
• Corollary 1.7
• Proposition 1.10
• Lemma 2.1: Fatou's Theorem for anti-rational maps
• remark 1
• Lemma 2.2: The Generalized Argument Principle for harmonic maps