Real Reparametrizations of Real Curves

In this paper we study the following two problems: first, given a rational parametrization (z) = (p1(z),p2(z)) (z)2of a complex curve in 2, to determine algorithmically, if has an infinite number of real points (i.e. if the trace of in R2is a real curve). If this is the case, then we would like to find another parametrization mapping of the same curve, but this time with real rational functions. The solution to both problems is given here by a simple algorithm, requiring essentially just a gcd computation and a parametrization of a real line or circle. On the other hand, the theoretical foundation for the algorithm seems more involved, relying on factorization properties of conjugate harmonic polynomials. The case of space curves or curves over a higher dimensional space follows by a direct generalization of our results or by considering the primitive element theorem.