Repeated differentiation of deterministic polynomials with asymptotically radial root distributions

2026/07/18 by Brian C. Hall, Daniel Perales
#math.PR #math-ph #math.MP #math.OA

paper · pdf

Abstract

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form P(z)=p(z^m), where p is a deterministic polynomial of degree n with real, non-negative roots, in the regime where m and n are large. If m≫ log(n) and the root distribution of P converges to a compactly supported, radial probability measure μ₀, these works show that for 0≤ t<1, the root distribution of the \lfloor nmt\rfloor-th derivative of P converges to a compactly supported probability measure μ_t given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator z^a(d/dz)^b. We also compute the limiting root distribution in the case when m is fixed and n tends to infinity.

Related

• Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators
• Asymptotic root distribution of polynomials under repeated polar differentiation
• Dynamics of zeroes under repeated differentiation
• Dynamics of rotationally invariant polynomial root sets under iterated differentiations
• The distribution of zeros of the derivative of a random polynomial