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 rectiable 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 rectication is proven indirectly, without an explicit construction of the mapping α. In this paper, we use Tarski-Seidenberg algorithm (for deciding the rst or-der theory of real numbers) to design an algorithm that, given a polynomial mapping φ: Ck → Cn which is known to be rectiable, returns a polynomial mapping α: Cn → Cn that recties 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. 1. Formulation of the Problem It is known that several classes of polynomial mappings are rectiable in the following sense. Denition 1. Let C denote the eld of all complex numbers. A polynomial mapping α: Cn → Cn is called a polynomial automorphism if this mapping a bijection, and the inverse mapping β = α1 is also polynomial. Denition 2. A polynomial mapping φ: Ck → Cn is called rectiable if these exists a polynomial automorphism α: Cn → Cn for which α(φ(t1,..., tk)) = (t1,..., tk, 0,...) for all (t1,..., tk). Most existing proofs of rectiability just prove the existence of a rectifying au-tomorphism α, without explaining how to actually compute it. In this paper, we show how to compute α. Copyrigh