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The number of real ovals of a cyclic cover of the sphere


Authors: Francisco-Javier Cirre and Peter Turbek
Journal: Proc. Amer. Math. Soc. 145 (2017), 2639-2647
MSC (2010): Primary 30F50; Secondary 14H30
DOI: https://doi.org/10.1090/proc/13411
Published electronically: December 15, 2016
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Abstract: A compact Riemann surface $ X$ which is a cyclic cover of degree $ n$ of the Riemann sphere has a defining equation of the form $ y^n=f(x)$ where $ f$ is a complex polynomial. If $ f$ has real coefficients, then complex conjugation $ \sigma $ leaves $ X$ invariant. The fixed point set of $ \sigma $ in $ X$ consists of a disjoint union of simple closed curves, called ovals. In this paper we determine a procedure to count the exact number of ovals of $ \sigma $ in terms of the multiplicities of the real roots of $ f.$


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Additional Information

Francisco-Javier Cirre
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educación a Distancia, 28040 Madrid, Spain
Email: jcirre@mat.uned.es

Peter Turbek
Affiliation: Department of Mathematics, Purdue University Northwest, 2200 169th Street, Hammond, Indiana 46323
Email: psturbek@pnw.edu

DOI: https://doi.org/10.1090/proc/13411
Received by editor(s): June 15, 2015
Received by editor(s) in revised form: July 22, 2016, and August 1, 2016
Published electronically: December 15, 2016
Additional Notes: The first author was partially supported by MTM2014-55812
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society