Let $\cal C$ be a space curve defined as the common zeros of $n-1$ polynomials. Assuming that $\cal C$ admits a rational parametrisation, there is an algorithm generating a point on the curve through algebraic operations, given a value $\tau$ on the base field. This mapping covers all the curve apart from a finite number of points. The input of the algorithm is given by the polynomials defining $\cal C$ and the parameter $\tau$.

What is the minimal number of required algebraic operations in terms of $n$ and the degree of the polynomials? Let me assume that this number does not depend on $\tau$.