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

preprint Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators Brian C. Hall author Ching-Wei Ho author Jonas Jalowy author 1137680000-0001-9624-2685 Zakhar Kabluchko author 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. 2023 eng arXiv:2312.14883 2312.14883 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. 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. 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., 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. @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., et al. “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, Z. Kabluchko, ArXiv:2312.14883 (2023). 592092025-03-31T07:15:40Z2025-04-23T14:38:56Z