## Analysis & PDE

• Anal. PDE
• Volume 12, Number 6 (2019), 1489-1512.

### 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 $Kn(l)(z)$ 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 behavior of zeros of $Kn(Nn)(z)$ depends on the limit of the ratio $Nn∕n$. In particular, we prove that when the limit of the ratio is strictly positive, the property of the uniform clustering around the unit circle fails; when the ratio is close to 1, the zeros have some rescaling phenomenon. Then we study such problem for random polynomials with more general coefficients. But things, especially the rescaling phenomenon, become very complicated for the general case when $Nn∕n→1$, where we compute the case of the random elliptic polynomials to illustrate this.

#### Article information

Source
Anal. PDE, Volume 12, Number 6 (2019), 1489-1512.

Dates
Revised: 22 August 2018
Accepted: 18 October 2018
First available in Project Euclid: 12 March 2019

https://projecteuclid.org/euclid.apde/1552356128

Digital Object Identifier
doi:10.2140/apde.2019.12.1489

Mathematical Reviews number (MathSciNet)
MR3921311

Zentralblatt MATH identifier
07061132

Subjects
Primary: 60E05: Distributions: general theory

#### Citation

Feng, Renjie; Yao, Dong. Zeros of repeated derivatives of random polynomials. Anal. PDE 12 (2019), no. 6, 1489--1512. doi:10.2140/apde.2019.12.1489. https://projecteuclid.org/euclid.apde/1552356128

#### References

• P. Bleher, B. Shiffman, and S. Zelditch, “Universality and scaling of correlations between zeros on complex manifolds”, Invent. Math. 142:2 (2000), 351–395.
• T. Bloom and B. Shiffman, “Zeros of random polynomials on $\mathbb C^m$”, Math. Res. Lett. 14:3 (2007), 469–479.
• D. W. Farmer and R. C. Rhoades, “Differentiation evens out zero spacings”, Trans. Amer. Math. Soc. 357:9 (2005), 3789–3811.
• D. W. Farmer and M. Yerrington, “Crystallization of random trigonometric polynomials”, J. Stat. Phys. 123:6 (2006), 1219–1230.
• R. Feng, “Zeros of derivatives of Gaussian random polynomials on plane domain”, preprint.
• J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series 51, American Mathematical Society, Providence, RI, 2009.
• I. Ibragimov and D. Zaporozhets, “On distribution of zeros of random polynomials in complex plane”, pp. 303–323 in Prokhorov and contemporary probability theory, edited by A. N. Shiryaev et al., Springer Proc. Math. Stat. 33, Springer, 2013.
• I. Ibragimov and O. Zeitouni, “On roots of random polynomials”, Trans. Amer. Math. Soc. 349:6 (1997), 2427–2441.
• Z. Kabluchko and D. Zaporozhets, “Roots of random polynomials whose coefficients have logarithmic tails”, Ann. Probab. 41:5 (2013), 3542–3581.
• Z. Kabluchko and D. Zaporozhets, “Asymptotic distribution of complex zeros of random analytic functions”, Ann. Probab. 42:4 (2014), 1374–1395.
• M. Kac, “On the average number of real roots of a random algebraic equation”, Bull. Amer. Math. Soc. 49 (1943), 314–320.
• L. A. Shepp and R. J. Vanderbei, “The complex zeros of random polynomials”, Trans. Amer. Math. Soc. 347:11 (1995), 4365–4384.
• B. Shiffman and S. Zelditch, “Equilibrium distribution of zeros of random polynomials”, Int. Math. Res. Not. 2003:1 (2003), 25–49.
• M. Sodin and B. Tsirelson, “Random complex zeroes, I: Asymptotic normality”, Israel J. Math. 144 (2004), 125–149.
• G. Szegő, Orthogonal polynomials, 4th ed., AMS Colloquium Publications 23, American Mathematical Society, Providence, R.I., 1975.