Some algebraic and geometric computations in PSPACE

  • Authors:

  • John Canny

  • Affiliations:

  • 543 Evans Hall, Computer Science Division, University of California, Berkley

  • Venue:
  • STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
  • Year:
  • 1988

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Abstract

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.