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Radical parametrizations of algebraic curves

A research project on computational algebraic geometry. Sponsored by the FWF (Austrian Science Fund).

Project number: P 22766-N18, approved on 27.09.2010.

Time period: 01.04.2011 to 31.03.2014 (duration: 36 months)

Associated institution: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences.

News

  • 23.02.2013 - Article First steps towards radical parametrization of algebraic surfaces accepted for publication in CAGD.
  • 03.10.2012 - Josef Schicho has become the project leader and David Sevilla has become an international research partner.
  • See older news below.

Members

  • David Sevilla (international research partner)
    • Phone: +34 924 289 300 ext 82557
    • Address:
      Centro Universitario de Mérida
      Avda. Santa Teresa de Jornet, 38
      E-06800 Mérida (Badajoz), Spain
  • Cevahir Demirkiran (former member)

Summary

It is well known that an algebraic curve can be parametrized by rational functions if and only if its genus is zero. The aim of this project is to study a more general class of curves: those parametrizable by radicals, that is, which admit a parametrization involving field operations and root extractions. This class is significantly larger than the previous one: it contains all elliptic and hyperelliptic functions, thus there are curves of every genus which have radical parametrizations.

We approach the problem from an algorithmical point of view. In other words, our ultimate goal would be to devise the best possible algorithms that decide whether a curve can be parametrized by radicals and compute one/all/the best radical parametrizations in the affirmative case. Naturally, this involves theoretical research into structural problems like a good definition of what is a better parametrization, what can we say about the structure of the class of all radical parametrization of a curve, etc. These questions go back to Zariski.

The proposal can be found here.

Publications

Peer-reviewed journals and monographs
  • J. R. Sendra, D. Sevilla: First steps towards radical parametrization of algebraic surfaces. CAGD, 2013. http://dx.doi.org/10.1016/j.cagd.2012.12.004
  • J. Schicho, D. Sevilla: Effective radical parametrization of trigonal curves. In “Computational Algebraic and Analytic Geometry”, Contemporary Mathematics, vol. 572, Amer. Math. Soc., Providence, RI, 2012, 221–231.
  • J. R. Sendra, D. Sevilla: Radical parametrizations of algebraic curves by adjoint curves. Journal of Symbolic Computation 46 (9), September 2011, 1030–1038. http://dx.doi.org/10.1016/j.jsc.2011.05.005
Conference proceedings
  • J. R. Sendra, D. Sevilla: First Steps Towards Radical Parametrization of Algebraic Surfaces. Accepted by the Eighth International Conference on Mathematical Methods for Curves and Surfaces (Oslo, June 28 – July 3, 2012).
Preprints
  • F.-O. Schreyer, J. Schicho, M. Weimann: Gonal maps and radical parametrizations of curves. Preprint.
  • J. R. Sendra, D. Sevilla: Some algebraic aspects of radical parametrizations of curves. Preprint.
  • H. Ahmadinezhad: Radical parametrization of non-rational algebraic surfaces. Preprint.

Further activities

Research visits (in and out)
  • Visit of D. Sevilla to J. R. Sendra at U. of Alcala in June 2012 to continue their work on algebraic aspects of radical parametrizations.
  • Visit of J. Schicho and M. Weimann to F.-O. Schreyer at U. Saarbrücken in May 2012 to work on algorithmic aspects of tetragonal curves and gonal maps.
  • Visit of J. R. Sendra to D. Sevilla at RICAM in February 2012 to work on algebraic aspects of radical parametrizations.
Participation in schools and workshops

Old news

  • 03.10.2012 - Josef Schicho has become the project leader and David Sevilla has become an international research partner.
  • 01.09.2012 - David Sevilla has relocated to the University of Extremadura in Spain.
  • 30.06.2012 - David Sevilla gave the talk First Steps Towards Radical Parametrization of Algebraic Surfaces in the Eighth International Conference on Mathematical Methods for Curves and Surfaces.
  • 12.06.2012 - Visit of David Sevilla to J. Rafael Sendra for one week.
  • 22.05.2012 - Visit of Josef Schicho, together with Martin Weimann, to Frank-Olaf Schreyer for one week.
  • 01.05.2012 - Hamid Admadinezhad is, as of today, a member of the project.
  • 28.03.2012 - Abstract First Steps Towards Radical Parametrization of Algebraic Surfaces submitted to the Eighth International Conference on Mathematical Methods for Curves and Surfaces (CAGD 2012) conference.
  • 02.03.2012 - Article Effective radical parametrization of trigonal curves accepted for publication in the Contemporary Math. Volume “Computational Algebraic and Analytic Geometry of Low-dimensional Varieties”.
  • 01.03.2012 - Hamid Admadinezhad, from Kent University, will join the project in the coming months.
  • 19.02.2012 - Visit of J. Rafael Sendra to David Sevilla for one week.
  • 15.02.2012 - Cevahir Demirkiran is no longer a member of the project, due to personal reasons.
  • 26.04.2011 - Article Effective radical parametrization of trigonal curves submitted.
  • 01.04.2011 - Start of the project.
  • 30.11.2010 - The vacancy has been filled.
  • 05.10.2010 - Official communication of project approval by the FWF. One Ph. D. position within the project is open for applications, see the vacancy page for details.
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