Variation of argument and Bernstein index for holomorphic functions on Riemann surfaces

• Autores: Yu Ilyashenko
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 14, Nº 3, 2007, págs. 433-442
• Idioma: inglés
• DOI: 10.4310/mrl.2007.v14.n3.a8
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• Resumen

  • An upper bound of the variation of argument of a holomorphic function along a curve on a Riemann surface is given. This bound is expressed through the Bernstein index of the function multiplied by a geometric constant. The Bernstein index characterizes growth of the function from a smaller domain to a larger one. The geometric constant in the estimate is explicitly given. This result is applied in \cite{GI1}, \cite{GI2} to the solution of the restricted version of the infinitesimal Hilbert 16th problem, namely, to upper estimates of the number of zeros of abelian integrals in complex domains.