Zero distributions of derivatives of polynomial families centering on a set

2023/12/22 by Henriksen, Christian, Petersen, Carsten Lunde, Uhre, Eva
#30C15 (Primary) 37F10 #42C05 (Secondary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.14519

Abstract

Suppose C ⊂ ℂ is compact. Let q_k be a sequence of polynomials of degree n_k → ∞, such that the locus of roots of all the polynomials is bounded, and the number of roots of q_k in any closed set L not meeting C is uniformly bounded. Supposing that (q_k)_k has an asymptotic root distribution μ we provide conditions on C and μ assuring the sequence of mth derivatives (q_k^(m))_k also has asymptotic root distribution μ for any m≥ 1. This complements recent results of Totik.

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