A cellular decomposition algorithm for semialgebraic sets

Abstract

For any r≥1 and any i, 0≤i≤r, an i-dimensional cell (in E^r) is a subset of r-dimensional Euclidean space E^r homeomorphic to the i-dimensional open unit ball. A subset of E^r 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 E^r) is the set of all points of E^r 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 E^r 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.

This research has been supported by NSF Grants MCS74-13278 Aol and MCS78-01731.