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).
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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.
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Journal Title
arXiv:2312.14883
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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.