Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

B.C. Hall, C.-W. Ho, J. Jalowy, Z. Kabluchko, ArXiv:2312.14883 (2023).

Download
No fulltext has been uploaded.
Preprint
Author
Hall, Brian C.; Ho, Ching-Wei; Jalowy, Jonas; Kabluchko, Zakhar
Abstract
We start with a random polynomial $P^{N}$ of degree $N$ with independent coefficients and consider a new polynomial $P_{t}^{N}$ obtained by repeated applications of a fraction differential operator of the form $z^{a}% (d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the limiting root distribution $\mu_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $\mu_{t}$ is the push-forward of the limiting root distribution of $P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing along the characteristic curves of the PDE satisfied by the log potential of $\mu_{t}.$ In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially \textit{with constant speed} until they hit the origin, at which point, they cease to exist. For general $a$ and $b,$ the transport map $T_{t}$ has a free probability interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal "transport operator." As an application, we obtain a push-forward characterization of the free self-convolution semigroup $\oplus$ of radial measures on $\mathbb{C}$. We also consider the case $b<0,$ which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.
Publishing Year
Journal Title
arXiv:2312.14883
LibreCat-ID

Cite this

Hall BC, Ho C-W, Jalowy J, Kabluchko Z. Roots of polynomials under repeated differentiation and repeated  applications of fractional differential operators. arXiv:231214883. Published online 2023.
Hall, B. C., Ho, C.-W., Jalowy, J., & Kabluchko, Z. (2023). Roots of polynomials under repeated differentiation and repeated  applications of fractional differential operators. In arXiv:2312.14883.
@article{Hall_Ho_Jalowy_Kabluchko_2023, title={Roots of polynomials under repeated differentiation and repeated  applications of fractional differential operators}, journal={arXiv:2312.14883}, author={Hall, Brian C. and Ho, Ching-Wei and Jalowy, Jonas and Kabluchko, Zakhar}, year={2023} }
Hall, Brian C., Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. “Roots of Polynomials under Repeated Differentiation and Repeated  Applications of Fractional Differential Operators.” ArXiv:2312.14883, 2023.
B. C. Hall, C.-W. Ho, J. Jalowy, and Z. Kabluchko, “Roots of polynomials under repeated differentiation and repeated  applications of fractional differential operators,” arXiv:2312.14883. 2023.
Hall, Brian C., et al. “Roots of Polynomials under Repeated Differentiation and Repeated  Applications of Fractional Differential Operators.” ArXiv:2312.14883, 2023.

Export

Marked Publications

Open Data LibreCat

Sources

arXiv 2312.14883

Search this title in

Google Scholar