Tracing index of rational curve parametrizations

A rational parametrization of an algebraic curve establishes a rational correspondence of this curve with the affine or projective line. This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational parametrization determines a linear tracing of the curve, when the parameter takes values in the algebraic closure of the ground field. Such a parametrization might trace the curve once or several times. We formally introduce the concept of the tracing index of a parametrization, we show how to compute it, and we relate it to the degree of rational reparametrizations as well as to the degree of the curve. In addition, we show how to apply these results to the case of real curves, where we introduce the notion of real tracing index.