We give a PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations. One of the consequences of this result is that the "Generalized Movers' Problem" in robotics drops from EXPTIME into PSPACE, and is therefore PSPACE-complete by a previous hardness result [Rei]. We also show that the existential theory of the real numbers can be decided in PSPACE. Other geometric problems that also drop into PSPACE include the 3-d Euclidean Shortest Path Problem, and the "2-d Asteroid Avoidance Problem" described in [RS]. Our method combines the theorem of the primitive element from classical algebra with a symbolic polynomial evaluation lemma from [BKR]. A decision problem involving several algebraic numbers is reduced to a problem involving a single algebraic number or primitive element, which rationally generates all the given algebraic numbers.
Title
Some Algebraic and Geometric Computations in PSPACE
Published
1988-08-01
Full Collection Name
Electrical Engineering & Computer Sciences Technical Reports
Other Identifiers
CSD-88-439
Type
Text
Extent
17 p
Archive
The Engineering Library
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