Zeros and Exponential Profiles of Polynomials I: Limit Distributions, Finite Free Convolutions and Repeated Differentiation
- 1Institute of Mathematics, Paderborn University, Warburger Str. 100, 33098, Paderborn, Germany
- 2Department of Mathematics and Computer Science, Institute of Mathematical Stochastics, University of Münster, Orléans-Ring 10, D-48149, Münster, Germany
- 3Department of Mathematics, Kyiv School of Economics, Kyiv, Ukraine
Abstract
Given a sequence of polynomials (Pn)n ∈ ℕ ( P n ) n ∈ N with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of Pn P n as deg Pn → ∞ deg P n → ∞ . The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution μ μ and the existence of an exponential profile g associated with the coefficients of the polynomials (Pn)n ∈ ℕ ( P n ) n ∈ N . The exponential profile g , which can be roughly described by [zk]Pn(z) ≈ exp (n g(k/n) ) [ z k ] P n ( z ) ≈ exp ( n g ( k / n ) ) , offers a direct route to computing the Cauchy transform G of μ μ : the functions t ↦ tG(t) t ↦ t G ( t ) and α ↦ exp (−g'(α ) ) α ↦ exp ( - g ′ ( α ) ) are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponent
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BibTeX
@article{jalowy2026zeros,
title = {{Zeros and Exponential Profiles of Polynomials I: Limit Distributions, Finite Free Convolutions and Repeated Differentiation}},
author = {Jonas Jalowy and Zakhar Kabluchko and Alexander Marynych},
journal = {Constructive Approximation},
year = {2026},
month = oct,
doi = {10.1007/s00365-026-09778-z},
url = {https://doi.org/10.1007/s00365-026-09778-z},
note = {CC BY},
}
RIS
TY - JOUR
TI - Zeros and Exponential Profiles of Polynomials I: Limit Distributions, Finite Free Convolutions and Repeated Differentiation
AU - Jonas Jalowy
AU - Zakhar Kabluchko
AU - Alexander Marynych
PY - 2026
DA - 2026/10/06
JO - Constructive Approximation
DO - 10.1007/s00365-026-09778-z
UR - https://doi.org/10.1007/s00365-026-09778-z
AB - Given a sequence of polynomials (P_n)_n ∈ ℕ ( P n ) n ∈ N with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of P_n P n as deg P_n → ∞ deg P n → ∞ . The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution μ μ and the existence of an exponential profile g associated with the coefficients of the polynomials (P_n)_n ∈ ℕ ( P n ) n ∈ N . The exponential profile g , which can be roughly described by [z^k]P_n(z) ≈ exp (n g(k/n) ) [ z k ] P n ( z ) ≈ exp ( n g ( k / n ) ) , offers a direct route to computing the Cauchy transform G of μ μ : the functions t ↦ tG(t) t ↦ t G ( t ) and α ↦ exp (−g'(α ) ) α ↦ exp ( - g ′ ( α ) ) are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponent
SN - 0176-4276
SN - 1432-0940
ER -