Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Some Algebraic and Geometric Computations in PSPACE

John F. Canny

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-88-439
August 1988

http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/CSD-88-439.pdf

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.


BibTeX citation:

@techreport{Canny:CSD-88-439,
    Author = {Canny, John F.},
    Title = {Some Algebraic and Geometric Computations in PSPACE},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1988},
    Month = {Aug},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/6041.html},
    Number = {UCB/CSD-88-439},
    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.}
}

EndNote citation:

%0 Report
%A Canny, John F.
%T Some Algebraic and Geometric Computations in PSPACE
%I EECS Department, University of California, Berkeley
%D 1988
%@ UCB/CSD-88-439
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/6041.html
%F Canny:CSD-88-439