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.

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of K-plane point-mass gravitational lens ensembles that produce K i=1 (5gi − 5) images of a single background source, where gi is the number of point masses in the i th plane. This gives asymptotically (for large gi with K fixed) 5 K times the minimal number of lensed images. Our construction uses Rhie's single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from differential topology, we then show that the preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including the positions of lensed images as well as the structure of the critical curves and caustics. We observe an interesting "caustic of multiplicity" phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case.