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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
MathSciNet review: 3667035
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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
Article copyright: © Copyright 2017 American Mathematical Society