What question did this study set out to answer?

To determine the behavior of derivatives of polynomial families regarding their root distributions as degrees increase.

March 8, 2026Open Access

Zero Distributions of Derivatives of Polynomial Families Centering on a Set

Key Points

To determine the behavior of derivatives of polynomial families regarding their root distributions as degrees increase.
Consider a compact set C in complex space.
Analyze polynomials q_k of degree n_k approaching infinity.
Establish conditions on C ensuring uniformly bounded roots in closed sets.
Examine asymptotic root distributions for the derivatives of polynomials.
Derivatives maintain the same asymptotic root distribution as their original polynomials.
Necessary conditions on C and root distributions μ are identified.

Abstract

Abstract Suppose C {C} C ⊂ C is compact. Let qₖ q k be a sequence of polynomials of degree nₖ n k → ∞, such that the locus of roots of all the polynomials is bounded, and the number of roots of qₖ q k in any closed set L disjoint from C is uniformly bounded. Supposing that (qₖ) ₖ (q k) k has an asymptotic root distribution μ we provide conditions on C and μ assuring the sequence of m th derivatives (qₖ^ (m) ) ₖ (q k (m) ) k has the same asymptotic root distribution μ for any m 1 m ≥ 1. This complements recent results of Totik, 19.

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