{"year":"2023","_id":"59209","publication":"arXiv:2312.14883","external_id":{"arxiv":["2312.14883"]},"date_updated":"2025-03-31T07:19:27Z","type":"preprint","title":"Roots of polynomials under repeated differentiation and repeated\n  applications of fractional differential operators","author":[{"full_name":"Hall, Brian C.","first_name":"Brian C.","last_name":"Hall"},{"full_name":"Ho, Ching-Wei","first_name":"Ching-Wei","last_name":"Ho"},{"full_name":"Jalowy, Jonas","first_name":"Jonas","last_name":"Jalowy"},{"full_name":"Kabluchko, Zakhar","first_name":"Zakhar","last_name":"Kabluchko"}],"date_created":"2025-03-31T07:15:40Z","citation":{"short":"B.C. Hall, C.-W. Ho, J. Jalowy, Z. Kabluchko, ArXiv:2312.14883 (2023).","bibtex":"@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} }","apa":"Hall, B. C., Ho, C.-W., Jalowy, J., &#38; Kabluchko, Z. (2023). Roots of polynomials under repeated differentiation and repeated   applications of fractional differential operators. In <i>arXiv:2312.14883</i>.","chicago":"Hall, Brian C., Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. “Roots of Polynomials under Repeated Differentiation and Repeated  Applications of Fractional Differential Operators.” <i>ArXiv:2312.14883</i>, 2023.","ama":"Hall BC, Ho C-W, Jalowy J, Kabluchko Z. Roots of polynomials under repeated differentiation and repeated   applications of fractional differential operators. <i>arXiv:231214883</i>. Published online 2023.","ieee":"B. C. Hall, C.-W. Ho, J. Jalowy, and Z. Kabluchko, “Roots of polynomials under repeated differentiation and repeated   applications of fractional differential operators,” <i>arXiv:2312.14883</i>. 2023.","mla":"Hall, Brian C., et al. “Roots of Polynomials under Repeated Differentiation and Repeated  Applications of Fractional Differential Operators.” <i>ArXiv:2312.14883</i>, 2023."},"status":"public","user_id":"113768","abstract":[{"lang":"eng","text":"We start with a random polynomial $P^{N}$ of degree $N$ with independent\ncoefficients and consider a new polynomial $P_{t}^{N}$ obtained by repeated\napplications of a fraction differential operator of the form $z^{a}%\n(d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the\nlimiting root distribution $\\mu_{t}$ of $P_{t}^{N}$ as $N\\rightarrow\\infty.$ We\nshow that $\\mu_{t}$ is the push-forward of the limiting root distribution of\n$P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing\nalong the characteristic curves of the PDE satisfied by the log potential of\n$\\mu_{t}.$ In the special case of repeated differentiation, our results may be\ninterpreted as saying that the roots evolve radially \\textit{with constant\nspeed} until they hit the origin, at which point, they cease to exist. For\ngeneral $a$ and $b,$ the transport map $T_{t}$ has a free probability\ninterpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal\n\"transport operator.\" As an application, we obtain a push-forward\ncharacterization of the free self-convolution semigroup $\\oplus$ of radial\nmeasures on $\\mathbb{C}$.\n  We also consider the case $b<0,$ which includes the case of repeated\nintegration. More complicated behavior of the roots can occur in this case."}]}