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A nonlocal transport equation modeling complex roots of polynomials under differentiation. (English) Zbl 1480.35343

For a random complex polynomial \(p_n\colon \mathbb{C}\to\mathbb{C}\) of degree \(n\) whose roots are distributed according to a radial density function \(u(|z|)dz\) on \(\mathbb{C}\), assuming that the limiting dynamics exists and roots remain separated, the authors deduce a mean field equation. They conjecture models for the density of roots at distance \(x\) at time \(t\) via the nonlocal transport equation \[ \frac{\partial\varphi}{\partial t}=\frac{\partial}{\partial x}\left(\left(\frac{1}{x}\int\limits_0^x\varphi(y)dy\right)^{-1}\varphi(x)\right). \] The main result is that an explicit solution \(\varphi(t,x)=\chi_{0\leq x\leq 1-t}\), which corresponds to the dynamics of random Taylor polynomials, has linear stability for small perturbations close to the origin. Here \(\varphi(t,r)\) denotes the distribution of roots after \(t\cdot n\)-times differentiation and \(\varphi(r)\) is the initial distribution. The linear stability of \(\varphi(t,x)\) is proven using the Hardy-type inequality.
In addition, the authors point out some open problems and discuss the possible ways to solve them.

MSC:

35Q49 Transport equations
44A15 Special integral transforms (Legendre, Hilbert, etc.)
82C70 Transport processes in time-dependent statistical mechanics
26C10 Real polynomials: location of zeros
31A99 Two-dimensional potential theory
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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