English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We start with a random polynomial $P^{N}$ of degree $N$ with independentcoefficients and consider a new polynomial $P_{t}^{N}$ obtained by repeatedapplications 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 rootdistribution $\mu_{t}$ 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 thecharacteristic curves of the PDE satisfied by the log potential of $\mu_{t}.$In the special case of repeated differentiation, our results may be interpretedas saying that the roots evolve radially with constant speed until they hit theorigin, at which point, they cease to exist.  As an application, we obtain a push-forward characterization of the freeself-convolution semigroup $\oplus$ of isotropic measures on $\mathbb C$. Wealso consider the case $b<0,$ which includes the case of repeated integration.More complicated behavior of the roots can occur in this case.

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