An algorithmic criterion for basicness in dimension 2
Abstract
We give a constructive procedure to check basicness of open (or closed) semialgebraic sets in a compact, non singular, real algebraic surface $X$. It is rather clear that if a semialgebraic set $S$ can be separated from each connected component of $X\setminus(S\cup\frz S)$ (when $\frz S$ stands for the Zariski closure of $(\ol S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$), then $S$ is basic. This leads to associate to $S$ a finite family of sign distributions on $X\setminus\frz S$; we prove the equivalence between basicness and two properties of these distributions, which can be tested by an algorithm. There is a close relation between these two properties and the behaviour of fans in the algebraic functions field of $X$ associated to a real prime divisor, which gives an easy proof, for a general surface $X$, of the well known 4elements fan's criterion for basicness (Brocker, AndradasRuiz). Furthermore, if the criterion fails, using the description of fans in dimension 2, we find an algorithmic method to exhibit the failure. Finally, exploiting this thecnics of sign distribution we give one improvement of the 4elements fan's criterion of Brocker to check if a semialgebraic set is principal.
 Publication:

arXiv eprints
 Pub Date:
 December 1993
 DOI:
 10.48550/arXiv.alggeom/9312006
 arXiv:
 arXiv:alggeom/9312006
 Bibcode:
 1993alg.geom.12006A
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 23 pages, amslatex (+bezier.sty) report: 1.89.(766) october 1993