Divisors and distribution of critical points

par Carsten Lunde Petersen

vendredi 2 oct. 2026, 10:30 → 12:30 Europe/Paris

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There are many results on the statistical properties of the root locus of various families of polynomials. Examples are:

Brolin’s theorem for the family of iterates of any fixed polynomial of degree at least $2$.

Stahl and Totiks theorem on the sequence of monic polynomials orthogonal with respect to a fixed probabibility meaure $\mu$ on $\mathbb C$ with compact non-polar support.

A common feature of these and several other examples is that the zeros of the polynomials in question are scarce, i.e. do not aggregate outside the filled-in support or polynomial convex hull of the support of $\mu$.

Recently attention has shifted to the much broader case of sequences of polynomials $q_k$, where the root distribution converges to a limit probability measure $\nu$ with compact support.

Assuming this condition I shall address the question of the distribution of critical points, i.e. the zeros of the derivatives $q’_k$ in areas where the roots of the polynomials $q_k$ are scarce.