i. Introduction In his 1948 paper A Decision Method for

inherently sequential. hand, purely algebraic.
Our method is, on the other We return to Tarski's
implemented
exponential time.
original formulation, and obtain parallelization by exploiting relying the algebraic on structure advances to the limit, in parallel algebra
notwithstanding, since that time.
that worst-case
give simpler sequential currently
algorithms
than the
in use, notably the cylindric algebraic method of Collins that [C]. seem t o CAD be
list of all consistent there are
exponentially if
possible
2
n-i "''qn
gcd(p,p' . . . . assignments, IIEII s n then at most 2n2+i of them We show how to do this in NC and
given independently by Collins and by Monk/Solovay. Various improvements and heuristics bound has stood
VLSI,
given the
current
however,
mathematical
ideas behind
procedure numbers
the +,
first-order ", and =,
has a real solution.
It is known how to solve the
real
problem sequentially in polynomial time [C].
commonly known as the theory of real closed fields. His decision procedure was sense). nonelementary (in the An elementary The practical importance of these problems is
A
straightforward growth
divide-and-conquer in degree and
yields
nonpolynomial The system is said
to
coefficient
be consistent
if it has a are at
size, so care must be taken. the parallel
for
determine univariate
whether
rational,
Elementary gave theory a
Algebra
Geometry," for with
Alfred
Tarski
polynomial
equations
and
inequalities
decision of the
with at most n bits. in NC (that i s , a
We will circuit
give an of depth a In
of all is a
and size n O(1) ) to determine with 1)Eli ~ n is
whether
such i.
(E,o)
and
,p(i)) = qi+lqi+2
are consistent.
458
gcd(p,p', .... p(i-1))'gcd(p,P', qi = gcd(p,p',...,p(i)) 2
and strict inequali£ies
for ~.
This step is not essential,
substantially.
I p(x)
< 0, if o(p) = -I = 0, if o(p) = 0 > 0, if 0(p) = 1 .
it does simplify the presentation
constraints
with ])~ & n, we show how to produce Suppose we are given a finite set ~ of rational univariate polynomials 0:E ~ {-l,O,l} p(x), and a sign assignment the system of equations refinement, that is, a new set of
THE COMPLEXITY OF ELEMENTARY ALGEBRA AND GEOMETRY (preliminary version)
Michael Ben-Or, MIT Dexter Kozen, IBM Research John Reif, Harvard In memoriam Alfred Tarski 1902-1983
consistent.
multiplicity
Thus
we know of no NC circuit that finds a real
2 n
solution of (~,o), even if we know that one exists; however, we can test consistency, there a r e . and even say how
further.
simple 2. Testing consistency of univariate polynomial
(multiplicity
polynomials are simple
E is called simple if the elements of and pairwise relatively prime. Given Z
P = qlq2 "''qn
many solutions
If 8 is a root of p of multiplicity root of p' of multiplicity m - i.
m, then 8 is a for i S m, m - i. Thus In
Thus
For
the
real
complexity-theoretic
cannot be overstated. crucial in symbolic
Polynomial manipulation manipulation and
decision procedure was given by Leonard Monk in his Berkeley PhD thesis, and in procedures 1974, were
in NC a simple polynomials F
such that F is simple, is a product assignment assignment for
[[r[[ S n21og(n),
and each p E A sign a sign but
representing
of powers of elements of F. r will uniquely determine
heavily for
recent and
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a generalized
Sturm
sequence
method,
and The
give basic
an
efficient behind but our
space log(n) 0(I).
parallel this
implementation. appears in
idea
method
Tarski's requires
closed
field
algorithm,
we will
8
is
a
root