THE THEOREM OF JENTZSCH–SZEGŐ ON AN ANALYTIC CURVE: APPLICATION TO THE IRREDUCIBILITY OF TRUNCATIONS OF POWER SERIES
Abstract
A theorem of Jentzsch–Szegő describes the limit measure of a sequence of discrete measures associated to zeroes of a sequence of polynomials in one variable. Following the presentation by Andrievskii and Blatt in [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] we extend this theorem to compact Riemann surfaces and to analytic curves in the sense of Berkovich over ultrametric fields, using classical potential theory in the former case, and Baker/Rumely, Thuillier's potential theory on analytic curves in the latter case. We then apply this equidistribution theorem to the question of irreducibility of truncations of power series with coefficients in ultrametric fields.
Résumé français: Le théorème de Jentzsch–Szegő décrit la mesure limite d'une suite de mesures discrètes associée aux zéros d'une suite convenable de polynômes en une variable. Suivant la présentation que font Andrievskii et Blatt dans [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] on étend ici ce résultat aux surfaces de Riemann compactes, puis aux courbes analytiques sur un corps ultramétrique. On donne pour finir quelques corollaires du cas particulier de la droite projective sur un corps ultramétrique à l'irréductibilité des polynômes-sections d'une série entière en une variable.
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