A cellular decomposition algorithm for semialgebraic sets
For any r≥1 and any i, 0≤i≤r, an i-dimensional cell (in Er) is a subset of r-dimensional Euclidean space Er homeomorphic to the i-dimensional open unit ball. A subset of Er is said to possess a cellular decomposition (c.d.) if it is the disjoint union of finitely many cells (of various dimensions). A semialgebraic set S (in Er) is the set of all points of Er satisfying some given finite boolean combination φ of polynomial equations and inequalities in r variables. φ is called a defining formula for S. A real algebraic variety, i.e. the set of zeros in Er of a system of polynomial equations in r variables, is a particular example of a semialgebraic set. It has been known for at least fifty years that any semialgebraic set possesses a c.d., but the proofs of this fact have been nonconstructive. Recently it has been noted that G. E. Collins' 1973 quantifier elimination algorithm for the elementary theory of real closed fields contains an algorithm for determining a c.d. of a semialgebraic set S given by its defining formula, apparently the first such algorithm. Specifically, each cell c of the c.d. C of S is itself a semialgebraic set, and for every c in C, a defining formula for c and a particular point of c are produced. In the present paper we provide a proof of this fact, our proof amounting to a description of Collins' algorithm from a theoretical point of view. We then show that the algorithm can be extended to determine the dimension of each cell in a c.d. and the incidences among cells. A computer implementation of the algorithm is in progress.
KeywordsAtomic Formula Quantifier Elimination Real Algebraic Variety Cellular Decomposition Real Closed Field
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- Collins, G. E., Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, 2nd GI Conf. on Automato Theory and Formal Lang., Lect. Notes in Comp. Sci. 33, Springer Verlag, Berlin, 1975, p. 134–183.Google Scholar
- Collins, G. E., Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition-A Synopsis, SIGSAM Bulletin of the Assoc. Comput. Mach., 10, 1 (1976), p. 10–12.Google Scholar
- Hironaka, H., Triangulations of Algebraic Sets, Proc. Symposia in Pure Math., 29, American Mathematical Society, Providence, 1975, p. 165–185.Google Scholar
- Kahn, P., private communication to G. Collins, 1978.Google Scholar
- Mathlab Group, MACSYMA Reference Manual, Version Nine, Laboratory for Computer Science, Mass. Inst. of Tech., 1977.Google Scholar
- Waerden, B. L. van der, Topologische Begründung des Kalküls der abzählenden Geometrie, Math. Ann. 102 (1929), p. 337–362.Google Scholar
- Waerden, B. L. van der, Modern Algebra v. II, tr. from the second revised German edn., Frederick Ungar, New York, 1950.Google Scholar
- Yun, David Y. Y., On Algorithms for Solving Systems of Polynomial Equations, SIGSAM Bulletin of the Assoc. Comput. Mach., #27 (Sept. 1973), p. 19–25.Google Scholar
- Arnon, D., Technical Report #353, Computer Sciences Dept., University of Wisconsin-Madison, 1979.Google Scholar