Computing roadmaps of semi-algebraic sets

on a variety

Authors:
Saugata Basu, Richard Pollack and Marie-Françoise Roy

Journal:
J. Amer. Math. Soc. **13** (2000), 55-82

MSC (1991):
Primary 14P10, 68Q25; Secondary 68Q40

DOI:
https://doi.org/10.1090/S0894-0347-99-00311-2

Published electronically:
July 20, 1999

MathSciNet review:
1685780

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a semi-algebraic set defined by polynomials in variables which is contained in an algebraic variety . The variety is assumed to have real dimension the polynomial and the polynomials defining have degree at most . We present an algorithm which constructs a *roadmap* on . The complexity of this algorithm is . We also present an algorithm which, given a point of defined by polynomials of degree at most , constructs a path joining this point to the roadmap. The complexity of this algorithm is These algorithms easily yield an algorithm which, given two points of defined by polynomials of degree at most , decides whether or not these two points of lie in the same semi-algebraically connected component of and if they do computes a semi-algebraic path in connecting the two points.

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Additional Information

**Saugata Basu**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
saugata@math.lsa.umich.edu

**Richard Pollack**

Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Email:
pollack@cims.nyu.edu

**Marie-Françoise Roy**

Affiliation:
IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu 35042 Rennes cedex, France

Email:
mfroy@univ-rennes1.fr

DOI:
https://doi.org/10.1090/S0894-0347-99-00311-2

Keywords:
Roadmaps,
semi-algebraic sets,
variety

Received by editor(s):
November 25, 1997

Received by editor(s) in revised form:
April 14, 1999

Published electronically:
July 20, 1999

Additional Notes:
The first author was supported in part by NSF Grants CCR-9402640 and CCR-9424398.

The second author was supported in part by NSF Grants CCR-9402640, CCR-9424398, DMS-9400293, CCR-9711240 and CCR-9732101.

The third author was supported in part by the project ESPRIT-LTR 21024 FRISCO and by European Community contract CHRX-CT94-0506.

Article copyright:
© Copyright 1999
American Mathematical Society