2019
DOI: 10.2140/apde.2019.12.1489
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Zeros of repeated derivatives of random polynomials

Abstract: It has been shown that zeros of Kac polynomials Kn(z) of degree n cluster asymptotically near the unit circle as n → ∞ under some assumptions. This property remains unchanged for the l-th derivative of the Kac polynomials K (l) n (z) for any fixed order l. So it's natural to study the situation when the number of the derivatives we take depends on n, i.e., l = Nn. We will show that the limiting global behavior of zeros of K (Nn) n (z) depends on the limit of the ratio Nn/n. In particular, we prove that when th… Show more

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Cited by 11 publications
(10 citation statements)
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“…There seem to be no results on this general PDE, but in the case when the initial distribution µ is rotationally invariant, the PDE simplifies considerably. The PDE corresponding to this isotropic special case has been first non-rigorously derived by O'Rourke and Steinerberger [42] and can be solved explicitly, as has been shown in [29]; see also [20] for a particular case when µ is the uniform distribution on the unit circle.…”
Section: Resultsmentioning
confidence: 99%
“…There seem to be no results on this general PDE, but in the case when the initial distribution µ is rotationally invariant, the PDE simplifies considerably. The PDE corresponding to this isotropic special case has been first non-rigorously derived by O'Rourke and Steinerberger [42] and can be solved explicitly, as has been shown in [29]; see also [20] for a particular case when µ is the uniform distribution on the unit circle.…”
Section: Resultsmentioning
confidence: 99%
“…We provide the details of the argument since it will be needed in the following. We take some t ∈ [0, 1) and look at the [tn]-th derivative of G n as defined in (6):…”
Section: 2mentioning
confidence: 99%
“…Here, t ∈ [0, 1) stays fixed, and [x] denotes the integer part of x. This question has been raised and studied in the papers of Steinerberger [28], O'Rourke and Steinerberger [21] and Feng and Yao [6]; see also [11,30,31]. Assigning a weight 1/n to each zero of the [tn]-th derivative, one can construct a subprobability measure on C denoted by µ (n) t :=…”
mentioning
confidence: 99%
“…In particular, do they obey a regular spacing at a local scale? We refer to [24,25,26,49] and references therein.…”
Section: Open Problemsmentioning
confidence: 99%
“…A result commonly attributed to Riesz [56] implies that the minimum gap between consecutive roots of p n is bigger than that of p n : zeroes even out and become more regular. We refer to results of Farmer & Rhoades [24], Farmer & Yerrington [25], Feng & Yao [26] and Pemantle & Subramnian [49]. Our result is inspired by a one-dimensional investigation due to the second author [55]:…”
mentioning
confidence: 92%