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

Cirre, Francisco-Javier; Turbek, Peter

doi:10.1090/proc/13411

The number of real ovals of a cyclic cover of the sphere
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by Francisco-Javier Cirre and Peter Turbek PDF

Proc. Amer. Math. Soc. 145 (2017), 2639-2647 Request permission

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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