The coefficient body of Bell representations of finitely connected planar domains

Abstract

In this paper, we determine the homotopy type of the coefficient body of Bell representations of non-degenerate n-connected planar domains with n 3. Also, by considering the isomorphism classes of rational functions, we get the precise number of those corresponding to Bell representations with same set of critical values. Further, the case of those with the same set of critical points is discussed.

Quadrature domains and kernel function zipping

“…I had conjectured in [6] that every n-connected domain in the plane such that no boundary component is a point is conformally equivalent to a domain with algebraic kernel functions. Jeong and Taniguchi [14] recently verified this conjecture. Since Gustafsson proved in [12] that every such domain is conformally equivalent to a quadrature domain of finite area, Theorem 1.2 gives an alternate way of seeing that every n-connected domain in the plane such that no boundary component is a point is conformally equivalent to a domain with algebraic kernel functions.…”

Equivalence problem for annuli and Bell representations in the plane

“…, b n−1 ) and some positive number r. Then he asked if every non-degenerate n-connected planar domain with n > 1 can be mapped biholomorphically onto the domain W a,b,r . Jeong and Taniguchi showed in [7] that it is true with r = 1. For every n 2 let B n be the set of (a, b) ∈ C 2n−2 such that the corresponding domains W a,b,1 are non-degenerate n-connected planar domains.…”

“…We call B n the coefficient body for non-degenerate n-connected canonical planar domains. The coefficient body B n is completely characterized by the same authors in [9]. W a,b, 1 are new canonical planar domains of connectivity n, which are called Bell representations.…”

On the geometry of random lemniscates

“…Lemniscates appear in a variety of specific studies and applications including approximation theory (e.g. Hilbert's lemniscate theorem and its generalizations [49,39]), topology of real algebraic curves [3,9,27], elliptic integrals from classical mechanics [2], holomorphic dynamics [38, p. 151], numerical analysis [46], operator theory [47], so-called "fingerprints" of twodimensional shapes [10,51], moving boundary problems [29,36], as critical sets of planar harmonic mappings [28,35], and in the theory and applications of conformal mapping [4,22,26].…”