#### Title

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

#### Publication Date

4-2015

#### Abstract

It is known that some polynomial mappings φ: C^{k} --> C^{n} are *rectifiable* in the sense that there exists a polynomial mapping α: C^{n} --> C^{n} whose inverse is also polynomial and for which α(φ(z_{1}, ...,z_{k})) = (z_{1}, ...,z_{k}, 0, ..., 0) for all z_{1}, ...,z_{k}. In many cases, the existence of such a rectification is proven indirectly, without an explicit construction of the mapping α.

In this paper, we use Tarski-Seidenberg algorithm (for deciding the first order theory of real numbers) to design an algorithm that, given a polynomial mapping φ: C^{k} --> C^{n} which is known to be rectifiable, returns a polynomial mapping α: C^{n} --> C^{n} that rectifies φ.

The above general algorithm is not practical for large n, since its computation time grows faster than 2^{(2n)}. To make computations more practically useful, for several important case, we have also designed a much faster alternative algorithm.

## Comments

Technical Report: UTEP-CS-15-31