Algebraically Optimal Parametrization

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.