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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 alldoi:10.1016/j.jsc.2007.09.002 fatcat:br54vbbgtjgebj3kd2zytybxii