Once We Know That a Polynomial Mapping Is Rectifiable, We Can Algorithmically Find a Rectification

It is known that some polynomial mappings φ : Ck → Cn are recti able in the sense that there exists a polynomial mapping α : Cn → Cn whose inverse is also polynomial and for which α(φ(z1, . . . , zk)) = (z1, . . . , zk, 0, . . . , 0) for all z1, . . . , zk. In many cases, the existence of such a recti cation is proven indirectly, without an explicit construction of the mapping α. In this paper, we use Tarski-Seidenberg algorithm (for deciding the rst order theory of real numbers) to design an algorithm that, given a polynomial mapping φ : Ck → Cn which is known to be recti able, returns a polynomial mapping α : Cn → Cn that recti es φ. The above general algorithm is not practical for large n, since its computation time grows faster than 22 n . To make computations more practically useful, for several important case, we have also designed a much faster alternative algorithm.