Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators
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arXiv
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| Format: | Preprint |
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2023
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| author | Hall, Brian C. Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar |
| author_facet | Hall, Brian C. Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar |
| contents | We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional 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 $μ_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $μ_{t}$ 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 a PDE satisfied by the log potential of $μ_{t}.$
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 \textquotedblleft transport operator.\textquotedblright 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14883 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators Hall, Brian C. Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar Probability 30C15 (Primary) 35F21, 60B20, 30C10, 60G57, 31A05, 60B10, 30D20, 46L54 (Secondary) We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional 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 $μ_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $μ_{t}$ 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 a PDE satisfied by the log potential of $μ_{t}.$ 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 \textquotedblleft transport operator.\textquotedblright 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. |
| title | Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators |
| topic | Probability 30C15 (Primary) 35F21, 60B20, 30C10, 60G57, 31A05, 60B10, 30D20, 46L54 (Secondary) |
| url | https://arxiv.org/abs/2312.14883 |