Conformal Models and Fingerprints of Pseudo-lemniscates

Abstract

Abstract. We prove that every meromorphic function on the closure of an analytic Jordan domain which is sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson and Shapiro by characterizing fingerprints of polynomial pseudolemniscates.The starting point of this paper is the following conjecture on the behavior of holomorphic functions on the closed unit disk.Conjecture 1. Let f be holomorphic on a neighborhood of the closure of the unit disk D. Then there is an injective holomorphic function φ : D → C and a polynomialIn this case, we say that f and p are conformally equivalent and that p is a conformal model for f on D. Note that D can be replaced by any Jordan domain, by the Riemann mapping theorem.The above was conjectured by the first author in [9], motivated by questions arising from the study of level curve configurations of meromorphic functions. The case where f is a finite Blaschke product follows from the work of Ebenfelft, Khavinson and Shapiro [1] on the characterization of the so-called fingerprints of polynomial lemniscates. Proofs of this special case were also given independently by the first author in [9] using level curves, and by the second author in [11] using a simple Riemann surfaces welding argument combined with the uniformization theorem.Conjecture 1 was proved in full generality by George Lowther and David Speyer on the internet mathematics forum math.stackexchange.com, using Lagrange interpolation and the inverse mapping theorem (see [8]). Our main motivation for this paper comes from the observation that the proof in [11] of the finite Blaschke product case can easily be generalized to obtain a proof of Conjecture 1, with some additional assumptions on the behavior of the function f on ∂D. Although this approach yields a slightly weaker result than that of Lowther and Speyer, it has considerable advantages. For instance, the argument also works in the meromorphic setting, in which case the polynomial p must be replaced by a rational map Date: February 24 2015.

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

“…In 2016, Richards [20] extended this result to general analytic functions which are analytic across the boundary of the unit disk, though this time with no control on the degree of the polynomial. In 2017, T. J. Richards and M. Younsi [23] gave a version of Theorem 4.4 for meromorphic functions, in which they were also able to recover control over the degree of the polynomial p (now a rational function q), subject to a condition on the behavior of the function f on the boundary of the disk. Theorem 4.5.…”

Conformal Equivalence of Analytic Functions on Compact Sets

“…Recently Younsi [6] (working jointly with the author) has applied these conformal welding techniques to provide a positive answer to the conformal modeling question for meromorphic functions on psuedo-tracts. That is, it has been shown that if D is a Jordan domain with smooth boundary, and f is meromorphic on cl(D), and f (∂D) is a Jordan curve, then f may be conformally modeled by a rational function r on D. One of the major advantages of this approach is that in the case described above the rational function r may be taken to have the smallest degree possible (namely the maximal number of preimages under f of any point, counted with multiplicity).…”

Computing Polynomial Conformal Models for Low-Degree Blaschke Products

“…As far as we know, the most general version is a solution on the online mathematics forum math.stackexchange.com by Lowther and Speyer [4,9] showing that the PCMQ has a positive answer as long as the domain D is bounded and the function f is analytic on the closure of D. This solution relies on approximation by polynomials interpolating certain derivative data, and can readily be generalized to meromorphic functions f , in which case polynomials need to be replaced by rational maps. Also of interest on this topic is the paper of the authors [8] which again brings the tools of conformal welding to bear on the PCMQ, also addressing the question of the degree of the polynomial conformal model in more detail.…”