Roots of polynomial sequences in root-sparse regions

2025/01/09 by Henriksen, Christian, Petersen, Carsten Lunde, Uhre, Eva
#31A15 #37F10 #42C05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.05203

Abstract

Given a family (q_k)_k of polynomials, we call an open set U root-sparse if the number of zeros of q_k is locally uniformly bounded on U. We study the interplay between the individual zeros of the polynomials q_k and those of the mth derivatives q_k^(m), in a root-sparse open set U, as k→∞. More precisely, if the root distributions μ_k of q_k converge weak* to some compactly supported measure μ, whose potential is nowhere locally constant on a root-sparse open set U, then we link the roots of the mth derivative q_k^m, for an arbitrary m>0, to the roots of q_k and the critical points of the potential p_μ on compact subsets of U. We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the mth derivative of iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials.

Related

• Zero distributions of derivatives of polynomial families centering on a set
• The distribution of zeros of the derivative of a random polynomial
• Root structures of polynomials with sparse exponents
• Roots of random polynomials with coefficients having polynomial growth
• The convergence of a sequence of polynomials and the distribution of their zeros