Computing hypercircles by moving hyperplanes

Let K be a field of characteristic zero and let α be an algebraic element of degree n over K. Given a proper parametrization ψ of a rational curve C with coefficients in K(α), we present a new algorithm to compute the hypercircle associated to the parametrization ψ. As a consequence, we can decide if C is defined over K and, if not, we can compute the minimum field of definition of C containing K. The algorithm exploits the structure of the conjugate curves of C but avoids computing in the normal closure of K(α) over K.