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A Maple Package to Deal with the Birationality of Curves and Surfaces Parametrizations

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Maple in Mathematics Education and Research (MC 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1414))

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Abstract

In this paper we present the Maple package Luroth for dealing with the birationality of curves and surfaces parametrizations. The procedures in the package decide whether a given, either curve or surface, parametrization is injective by computing its degree map. In addition, if the parametrization is not injective, it determines a birational reparametrization. For the curve case, the corresponding command always provides an optional answer. For the surface case, not all cases are covered. Nevertheless, we illustrate using Maple some new ideas on how to approach those surface cases not covered in the package.

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References

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Acknowledgements

This work has been partially supported by FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación/MTM2017-88796-P (Symbolic Computation: new challenges in Algebra and Geometry together with its applications). Authors belong to the Research Group ASYNACS (Ref. CT-CE2019/683).

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Authors and Affiliations

  1. Universidad de Alcalá, Dpto. Física y Matemáticas, Alcalá de Henares, Madrid, Spain

    Jorge Caravantes, Sonia Pérez–Díaz & J. Rafael Sendra

Authors
  1. Jorge Caravantes
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  2. Sonia Pérez–Díaz
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  3. J. Rafael Sendra
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Corresponding author

Correspondence to Jorge Caravantes .

Editor information

Editors and Affiliations

  1. Western University, London, ON, Canada

    Robert M. Corless

  2. Maplesoft, Waterloo, ON, Canada

    Jürgen Gerhard

  3. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias S. Kotsireas

5 Appendix

5 Appendix

In this appendix, the Maple executions, corresponding to the examples in the Subsect. 3.2, are shown.

Fig. 1.
figure 1

Starting the package

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Fig. 2.
figure 2

It checks the properness of \(\mathcal P\) in Example 4. The same result is achieved with the option deterministic

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Fig. 3.
figure 3

It computes a proper parametrization of the curve in Example 4

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Fig. 4.
figure 4

It checks the properness of \(\mathcal P\) in Example 5. The same result is achieved with the option probabilistic. Applying the command SurfacePorperReparametrization one gets a proper parametrization of the surface.

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Fig. 5.
figure 5

It checks the properness of \(\mathcal P\) in Example 6. The same result is achieved with the option probabilistic. Applying the command SurfacePorperReparametrization one gets a degree map 4 parametrization of the surface.

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Fig. 6.
figure 6

It checks the properness of \(\mathcal P\) in Example 7. The same result is achieved with the option probabilistic. The command SurfacePorperReparametrization does not get any parametrization with smaller map degree.

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Caravantes, J., Pérez–Díaz, S., Sendra, J.R. (2021). A Maple Package to Deal with the Birationality of Curves and Surfaces Parametrizations. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_10

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  • DOI: https://doi.org/10.1007/978-3-030-81698-8_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-81697-1

  • Online ISBN: 978-3-030-81698-8

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