Skip to main content
SpringerLink
Log in

Classifying Cubic Surfaces over Finite Fields Using Orbiter

  • Conference paper
  • First Online:
Mathematical Software – ICMS 2018 (ICMS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10931))

Included in the following conference series:

Abstract

We present two algorithms to classify cubic surfaces over a finite fields. An implementation in the programming system Orbiter will be described.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Betten, A.: Classifying discrete objects with orbiter. ACM Commun. Comput. Algebra 47(3/4), 183–186 (2014). https://doi.org/10.1145/2576802.2576832

    Article  Google Scholar 

  2. Betten, A.: Orbiter - a program to classify discrete objects, 2016–2018. https://github.com/abetten/orbiter

  3. Betten, A.:. Rainbow cliques and the classification of small BLT-Sets. In: Kauers, M. (ed.) ISSAC 2013, 26–29 June 2013, Boston, Massachusetts, pp. 53–60 (2013)

    Google Scholar 

  4. Betten, A., Hirschfeld, J.W.P., Karaoglu, F.: Classification of cubic surfaces with twenty-seven lines over the finite field of order thirteen. Eur. J. Math. 4(1), 37–50 (2018)

    Article  MathSciNet  Google Scholar 

  5. Betten, A., Karaoglu, F.: Cubic surfaces over small finite fields. Submitted to Designs, Codes and Cryptography

    Google Scholar 

  6. Clebsch, A.: Die Geometrie auf den Flächen dritter Ordnung. J. Reine Angew. Math. 65, 359–380 (1866)

    Article  MathSciNet  Google Scholar 

  7. Esculier, A.: Clebsch, ses 27 droites, les points de Eckardt. http://aesculier.fr/fichiersMaple/ClebschDroites/ClebschDroites.html. Accessed 15 Apr 2018

  8. Hirschfeld, J.W.P.: Finite Projective Spaces of Three Dimensions, x+316 pp. Oxford University Press, Oxford (1985)

    Google Scholar 

  9. Karaoglu, F.: The cubic surfaces with twenty-seven lines over finite fields, Thesis, University of Sussex (2018). Submitted

    Google Scholar 

  10. Povray. Persistence of vision raytracer. http://www.povray.org

  11. Schläfli, L.: An attempt to determine the twenty-seven lines upon a surface of the third order and to divide such surfaces into species in reference to the reality of the lines upon the surface. Quart. J. Math. 2, 55–110 (1858)

    Google Scholar 

Download references

Acknowledgements

The author thanks Alain Esculier [7] for providing the original figure of the Clebsch surface with the accompanying Povray [10] source code. He also thanks Professor Hirschfeld and Fatma Karaoglu for stimulating discussions during a Sabbatical stay at the University of Sussex in the Fall of 2017.

Author information

Authors and Affiliations

  1. Colorado State University, Fort Collins, USA

    Anton Betten

Authors
  1. Anton Betten
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton Betten .

Editor information

Editors and Affiliations

  1. University of Bath, Bath, United Kingdom

    James H. Davenport

  2. Johannes Kepler University, Linz, Austria

    Manuel Kauers

  3. University of Waterloo, Waterloo, Ontario, Canada

    George Labahn

  4. Czech Technical University in Prague, Prague 6, Czech Republic

    Josef Urban

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Betten, A. (2018). Classifying Cubic Surfaces over Finite Fields Using Orbiter. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-96418-8_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96417-1

  • Online ISBN: 978-3-319-96418-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation