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A parametric version of the Hilbert-Hurwitz theorem using hypercircles

Author: Luis Felipe Tabera
Journal: Math. Comp. 86 (2017), 3001-3018
MSC (2010): Primary 14Q05, 68W30, 14M20
Published electronically: April 7, 2017
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Abstract: Let $ \mathbb{K}$ be a characteristic zero field, let $ \alpha $ be an algebraic element over $ \mathbb{K}$ and $ \mathcal {C}$ a rational curve defined over $ \mathbb{K}$ given by a parametrization $ \psi $ with coefficients in $ \mathbb{K}(\alpha )$. We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction $ u=\frac {at+b}{ct+d}$ such that $ \psi (u)$ has coefficients over an algebraic extension of $ \mathbb{K}$ of degree at most two and a conic $ \mathbb{K}$-birational to $ \mathcal {C}$. Moreover, if the degree of $ \mathcal {C}$ is odd or $ \alpha $ is of odd degree over $ \mathbb{K}$, we can compute a parametrization of $ \mathcal {C}$ with coefficients over $ \mathbb{K}$. The problem is solved without implicitization methods nor analyzing the singularities of  $ \mathcal {C}$.

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Additional Information

Luis Felipe Tabera
Affiliation: Departamento de Matemáticas Estadística y Computación, Universidad de Cantabria, 39071, Santander, Spain

Keywords: Hypercircles, rational curves, rational points
Received by editor(s): August 1, 2011
Received by editor(s) in revised form: February 19, 2014, and July 26, 2016
Published electronically: April 7, 2017
Additional Notes: The author is supported by the Spanish “Ministerio de Ciencia e Innovación” projects MTM2008-04699-C03-03 and MTM2011-25816-C02-02 and “Ministerio de Economa y Competitividad” and by the European Regional Development Fund (ERDF) project MTM2014-54141-P
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