Detecting real singularities of a space curve from a real rational parametrization

In this paper we give an algorithm that detects real singularities, including singularities at infinity, and counts local branches and multiplicities of real rational curves in the affine n-space without knowing an implicitization. The main idea behind this is a generalization of the D-resultant (see [van den Essen, A., Yu, J.-T., 1997. The D-resultant, singularities and the degree of unfaithfulness. Proc. Amer. Math. Soc. 25 (3), 689–695]) to n rational functions. This allows us to find all real parameters corresponding to the real singularities between the solutions of a system of polynomials in one variable.