Distribution asymptotique des valeurs propres des endomorphismes de Frobenius [d'apr\`es Abel, Chebyshev, Robinson, ...]

TL;DR

This paper investigates the asymptotic distribution of roots of unitary polynomials with roots in a compact set, analyzing the limiting measures and their supports, especially when the set is within the real numbers.

Contribution

It provides partial answers on the limiting behavior and support of root distributions for polynomials with roots in a given compact set, extending previous results.

Findings

Limits of root measures are characterized for polynomials with roots in compact sets.

Supports of limiting measures are described, especially for real sets.

Partial results are obtained for the case when roots lie in the real line.

Abstract

We consider unitary polynomials $P \in Z[X]$ whose roots $(x_i)$ belong to a given compact $K$ of $C$. To such a polynomial we associate the measure $\mu_P$ on $K$ which is the mean value of the Dirac measures $\delta_{x_i}$. What are the limits of the measures $\mu_P$ when $P$ varies ? In particular, what are their supports? We give partial answers to such questions, especially when $K$ is contained in $R$.