A nonlocal transport equation modeling complex roots of polynomials under differentiation

Let p n : C → p_n:\mathbb {C} \rightarrow \mathbb {C} be a random complex polynomial whose roots are sampled i.i.d. from radial distribution alttext="2 pi r u left-parenthesis right-parenthesis d r"> 2 π r u stretchy="false">( stretchy="false">) d encoding="application/x-tex">2\pi u(r) dr in the plane. A natural question is how of evolves under repeated (say alttext="n slash 2 minus"> / −<!-- − encoding="application/x-tex">n/2- times) differentiation polynomial. We conjecture mean-field expansion for evolution alttext="psi s equals s"> ψ<!-- ψ <mml:mi>s = encoding="application/x-tex">\psi (s) = u(s) s : mathvariant="normal">∂<!-- ∂ <mml:mi>t x ( 1 ∫<!-- ∫ <mml:mn>0 ) . encoding="application/x-tex">\begin{equation*} \frac {\partial \psi }{\partial t} x} \left ( {1}{x} \int _{0}^{x} ds \right )^{-1} (x) ). \end{equation*} The identical-to 1"> ≡<!-- ≡ \equiv 1</mml:annotation> corresponds to Taylor polynomials z sigma-summation Underscript k Overscript Endscripts gamma factorial where tilde script N comma z ∑<!-- ∑ <mml:mi>k γ<!-- γ <mml:mo>! where ∼<!-- ∼ class="MJX-tex-caligraphic" mathvariant="script">N , p_n(z) \sum _{k=0}^{n}{ \gamma _k {z^k}{k!}} \quad \text {where} \sim \mathcal {N}_{\mathbb {C}}(0,1). discuss some numerical examples suggesting that this particular solution may stable. prove linearly linear stability analysis reduces classical Hardy inequality. Many open problems discussed.