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)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 ≥ 1 the valence is at most n 2 +m 2 . This improves, for every choice of n, m ≥ 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.