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

Authors

Julio Urenda, The University of Texas at El PasoFollow
David Finston, New Mexico State UniversityFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

4-2015

Comments

Technical Report: UTEP-CS-15-31

Abstract

It is known that some polynomial mappings φ: C^k --> Cⁿ are rectifiable in the sense that there exists a polynomial mapping α: Cⁿ --> Cⁿ whose inverse is also polynomial and for which α(φ(z₁, ...,z_k)) = (z₁, ...,z_k, 0, ..., 0) for all z₁, ...,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 --> Cⁿ which is known to be rectifiable, returns a polynomial mapping α: Cⁿ --> Cⁿ 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.