The Theorem of Jentzsch--Szeg\H{o} on an analytic curve. Application to the irreducibility of truncations of power series

TL;DR

This paper extends the Jentzsch--Szeg ext{"o}"} theorem to analytic curves over various fields, providing new insights into the distribution of zeros of polynomials and implications for the irreducibility of power series truncations.

Contribution

It generalizes the classical theorem to compact Riemann surfaces and ultrametric fields, linking zero distribution to algebraic properties of power series.

Findings

Extended the theorem to Riemann surfaces and ultrametric fields

Provided criteria for irreducibility of power series truncations

Connected zero distribution with algebraic properties of polynomials

Abstract

The theorem of Jentzsch--Szeg\H{o} describes the limit measure of a sequence of discrete measures associated to the zeroes of a sequence of polynomials in one variable. Following the presentation of this result by Andrievskii and Blatt in their book, we extend this theorem to compact Riemann surfaces, then to analytic curves over an ultrametric field. The particular case of the projective line over an ultrametric field gives as corollaries information about the irreducibility of the truncations of a power series in one variable.