A PARAMETRIC VERSION OF THE HILBERT-HURWITZ THEOREM USING HYPERCIRCLES

Let K be a characteristic zero field, let a be an algebraic element over K and C a rational curve defined over K given by a parametrization (sic) with coefficients in K( a). We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction u = at+b/ct+d such that (sic) (u) has coefficients over an algebraic extension of K of degree at most two and a conic K-birational to C. Moreover, if the degree of C is odd or a is of odd degree over K, we can compute a parametrization of C with coefficients over K. The problem is solved without implicitization methods nor analyzing the singularities of C.