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

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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.

**1.**S. BASU, R. POLLACK, M.-F. ROY*On the combinatorial and algebraic complexity of Quantifier Elimination*,*J. Assoc. Comput. Machin.*, 43, 1002-1045, (1996). MR**98c:03077****2.**S. BASU, R. POLLACK, M.-F. ROY*Computing Roadmaps of Semi-algebraic Sets*,*Proc. 28th Annual ACM Symposium on the Theory of Computing*, 168-173, (1996). CMP**97:06****3.**S. BASU, R. POLLACK, M.-F. ROY*On Computing a Set of Points meeting every Semi-algebraically Connected Component of a Family of Polynomials on a Variety*, J. Complexity, Vol 13, Number 1, 28-37 (1997). MR**98d:14071****4.**S. BASU, R. POLLACK, M.-F. ROY*Computing Roadmaps of Semi-algebraic Sets on a Variety*,*Foundations of Computational Mathematics*, F. Cucker and M. Shub, Eds., 1-15, (1997). CMP**99:05****5.**J. BOCHNAK, M. COSTE, M.-F. ROY Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 36, Berlin: Springer-Verlag (1998). CMP**99:07****6.**J. CANNY, The Complexity of Robot Motion Planning, MIT Press (1987). MR**89m:68142****7.**J. CANNY,*Computing road maps in general semi-algebraic sets*, The Computer Journal, 36: 504-514, (1993). MR**94i:68108****8.**J. CANNY, D. GRIGOR'EV, N. VOROBJOV*Finding connected components of a semi-algebraic set in subexponential time,*Applic. Alg. Eng. Comput. 2: 217-238 (1992). MR**95m:03078****9.**G. E. COLLINS,*Quantifier elimination for real closed fields by cylindrical algebraic decomposition*, Autom. Theor. form. Lang., 2nd GI Conf., Kaiserslautern 1975, Lect. Notes Comput. Sci. 33, 134-183 (1975). MR**53:7771****10.**M. COSTE, M.-F. ROY*Thom's lemma, the coding of real algebraic numbers and the topology of semi-algebraic sets,*J. Symb. Comput. 5, No.1/2, 121-129 (1988). MR**89g:12002****11.**M. COSTE, M. SHIOTA*Nash triviality in families of Nash manifolds,*Invent. Math. 108, 349-368 (1992). MR**93e:14066****12.**D. GRIGOR'EV, N. VOROBJOV*Counting connected components of a semi-algebraic set in subexponential time*, Comput. Complexity 2, No.2, 133-186 (1992). MR**94a:68062****13.**L. GOURNAY, J. J. RISLER*Construction of roadmaps of semi-algebraic sets,*Appl. Algebra Eng. Commun. Comput. 4, No.4, 239-252 (1993). MR**94j:14049****14.**R. M. HARDT*Semi-algebraic Local Triviality in Semi-algebraic Mappings*, Am. J. Math. 102, 291-302 (1980). MR**81d:32012****15.**J. HEINTZ, M.-F. ROY, P. SOLERNÒ*Single exponential path finding in semi-algebraic sets I: The case of a regular hypersurface*, Discrete and Applied Math., Proc.AAECC-8 Tokyo, 1990; Lectures Notes in Comp. Sci. 508, Springer-Verlag 180-186 (1991). MR**92j:14072****16.**J. HEINTZ, M.-F. ROY, P. SOLERNÒ*Single exponential path finding in semi-algebraic sets II : The general case*, Bajaj, Chandrajit L. (ed.), Algebraic geometry and its applications. Collections of papers from Shreeram S. Abhyankar's 60th birthday conference held at Purdue University, West Lafayette, IN, USA, June 1-4, 1990. New York: Springer-Verlag, 449-465 (1994). MR**95i:14054****17.**J.-C. LATOMBE*Robot Motion Planning*, The Kluwer International Series in Engineering and Computer Science. 124. Dordrecht: Kluwer Academic Publishers Group (1991).**18.**J. MILNOR*Morse Theory*, Princeton: Princeton Univ. Press (1963). MR**29:634****19.**J. RENEGAR*On the computational complexity and geometry of the first order theory of the reals, parts I, II and III.*, J. Symb. Comput., 13 (3) 255-352, (1992). MR**93h:03011a**; MR**93h:03011b**; MR**93h:03011c****20.**F. ROUILLIER, M.-F. ROY, M. SAFEY*Finding at least a point in each connected component of a real algebraic set defined by a single equation,*to appear in Journal of Complexity.**21.**M.-F. ROY, N.N. VOROBJOV*Finding irreducible components of some real transcendental varieties*, Journal of Computationnal Complexity 4 107-132 (1994). MR**95g:68056****22.**M.-F. ROY, N. VOROBJOV*Computing the Complexification of a Semi-algebraic Set*, Proc. of International Symposium on Symbolic and Algebraic Computations, 1996, 26-34 (complete version to appear in Math. Zeitschrift).**23.**J. SCHWARTZ, M. SHARIR*On the `piano movers' problem II. General techniques for computing topological properties of real algebraic manifolds*, Adv. Appl. Math. 4, 298-351 (1983). MR**85h:52014**

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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