Proper real reparametrization of rational ruled surfaces

Let K⊆RK⊆R be a computable field. We present an algorithm to decide whether a proper rational parametrization of a ruled surface, with coefficients in K(i)K(i), can be properly reparametrized over a real (i.e. embedded in RR) finite field extension of KK. Moreover, in the affirmative case, the algorithm provides a proper parametrization with coefficients in a real extension of KK of degree at most 2.

Research highlights
► Given a complex (non-real) proper parametrization of a ruled surface, we provide an algorithm to reparametrize it (if possible) over the reals.