A new estimate of the transfinite diameter of Bernstein sets

2023/05/31 by Dimitri Jordan Kenne, Kenne, Dimitri Jordan
Engineering · Mathematics · #31C15 #32U15 #41A17 #Advanced Numerical Analysis Techniques #Complex Variables (math.CV) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2306.00216

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K ⊂ ℂⁿ be a compact set satisfying the following Bernstein inequality: for any m ∈ \ 1,..., n\ and for any n-variate polynomial P of degree deg(P) we have max_{z∈ K}|(∂ P)/(∂ z_m)(z)| ≤ M deg(P) max_{z∈ K}|P(z)| for z = (z₁, …, z_n). for some constant M= M(K)>0 depending only on K. We show that the transfinite diameter of K, denoted δ(K), verifies the following lower estimate δ(K) ≥ (1)/(n M), which is optimal in the one-dimensional case. In addition, we show that if K is a Cartesian product of compact planar sets then δ(K) ≥ (1)/(M).

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