A Thought on Approximation by Bi-Analytic Functions

Khavinson, Dmitry

doi:10.1007/978-1-4939-7543-3_6

Cited by 1 publication

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“…The above Problems 1.1 and 1.8 thus fit into a broader setting concerning the valence of polyanalytic polynomials. Polyanalytic polynomials have been studied classically [3] and have also arisen in more recent studies in gravitational lensing [33], [31], approximation theory [11], [8], [18] numerical linear algebra [15], random matrix theory [17], and determinantal point processes [2]. Motivated by the valence problems for harmonic and logharmonic polynomials as well as the prominent role played by zero sets in several of the aforementioned studies on polyanalytic polynomials, it seems natural to pose the following general valence problem for polyanalytic polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the valence of logharmonic polynomials

Khavinson¹,

Lundberg²,

Perry³

2023

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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.

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