REAL REPARAMETRIZATIONS OF REAL CURVES

In this paper we study the following two problems: first, given a rati onal parametrization P(z) = (p(1)(z), p(2)(z)) is an element of C(z)(2 ) of a complex curve C in C-2, to determine algorithmically, if C has an infinite number of real points (i.e. if the trace of C in R(2) is 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 rat ional functions. The solution to both problems is given here by a simp le algorithm, requiring essentially just a gcd computation and a param etrization of a real line or circle. On the other hand, the theoretica l foundation for the algorithm seems more involved, relying on factori zation 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. (C) 1997 Academic Press Limited.