On a question of J. P. Serre

Abstract: In his Bourbaki talk "{\em Distribution asymptotique des valeurs propres des endomorphismes de Frobenius (d'après Abel, Chebyshev, Robinson,$\cdots)$}" of 03/31/2018, J.P. Serre raised the following question:

Let $K\subset \BC$ be a compact (infinite) set, stable under complex conjugation and having a capacity greater than one. Let $U$ be an open set containing $K$. Then there exists a sequence of monic polynomials $P_n(X)\in \BZ[X]$, ${\rm deg} P_n= n,\; P_n^{-1}(0)\subset U$ and

\[\displaystyle \lim_{n\to \infty} \frac{1}{n} \sum_{P_n(z)=0} \delta_z= \mu \] where $\mu$ is the equilibrium measure of $K$ and $\delta_z$ is the Dirac measure at $z$. \\ I will present some results on this question, obtained with Th\'er\`ese Falliero(University of Avignon).

analysis of PDEsclassical analysis and ODEscomplex variablesdifferential geometrydynamical systemsfunctional analysismetric geometry

Audience: researchers in the topic

Analysis and Geometry Seminar