Advertisement

Algebraically Optimal Parametrization

Chapter
  • 1.3k Downloads
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 22)

In Chap. 4 we have analyzed the parametrization problem for rational curves, and we have presented algorithms for this purpose. Furthermore, we have proved that these algorithms determine proper parametrizations. Therefore, we can ensure that the parametrizations generated by these algorithms are optimal w.r.t. the degree of the components (see Theorem 4.21 and Corollary 4.22). In this chapter we analyze a different optimality criterion for parametrizations, namely the degree of the field extension necessary for representing coefficients of the parametrization. For instance, the parametrization (√2t, 2t2) of the parabola is optimal w.r.t. the degree (i.e., is proper) but it is expressed over ℚ(√2), while the alternative parametrization (t, t2) is expressed over ℚ and is also optimal w.r.t. the degree. Thus, we are interested in computing proper parametrizations that require the smallest possible field extension of the ground field. After introducing the notion of the field of parametrization in Section 5.1 and describing the Legendre method for finding rational points on conics in Section 5.2, we present in Section 5.3 an algebraically optimal parametrization of algebraic curves.

Keywords

Optimal Parametrization Rational Point Nontrivial Solution Double Point Conjugate Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Personalised recommendations