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St. Petersburg Mathematical Journal

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Separating semigroup of hyperelliptic curves and of genus 3 curves


Author: S. Yu. Orevkov
Translated by: S. Yu. Orevkov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 1.
Journal: St. Petersburg Math. J. 31 (2020), 81-84
MSC (2010): Primary 20G15
DOI: https://doi.org/10.1090/spmj/1586
Published electronically: December 3, 2019
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Abstract | References | Similar Articles | Additional Information

Abstract: A rational function on a real algebraic curve $ C$ is said to be separating if it takes real values only at real points. Such a function gives rise to a covering $ \mathbb{R} C\to \mathbb{R}\mathbb{P}^1$. Let $ A_1,\dots ,A_n$ be connected components of $ \mathbb{R} C$. In a recent paper, M. Kummer and K. Shaw defined the separating semigroup of $ C$ as the set of all sequences $ (d_1(f),\dots ,d_n(f))$ where $ f$ is a separating function and $ d_i$ is the degree of the restriction of $ f$ to $ A_i$.

Here, the separating semigroups for hyperelliptic curves and for genus 3 curves are described.


References [Enhancements On Off] (What's this?)

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Additional Information

S. Yu. Orevkov
Affiliation: Steklov Mathematical Institute, Gubkina 8, Moscow, Russia; IMT, l’université Paul Sabatier, 118 route de Narbonne, Toulouse, France
Email: orevkov@math.ups-tlse.fr

DOI: https://doi.org/10.1090/spmj/1586
Keywords: Real algebraic curve, curve of type I, dividing semigroup
Received by editor(s): December 11, 2017
Published electronically: December 3, 2019
Additional Notes: Partially supported by RFBR grant no. 17-01-00592-a
Article copyright: © Copyright 2019 American Mathematical Society