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110 CHAPTER 3. POLYNOMIAL EQUATIONS
3.5.6 Complexity
Let us suppose that there are s polynomials involved in the input formula (3.54),
of maximal degree d. ( Then such a cylindrical algebraic decomposition can be
computed in time O (sd) 2O(k))) .
There are examples [BD07, DH88], which shows that this behaviour is bestpossible,
indeed the projection onto R 1 might have a number of components
doubly-exponential in k.
While this behaviour is intrinsic to cylindrical algebraic decomposition, it is
not necessarily intrinsic to quantifier elimination as such. If a is the number of
alternations of quantifiers (Notation 17) in the problem (so a < k), then there
are algorithms [Bas99, for example] whose ( behaviour is singly-exponential in k
but doubly-exponential in a; typically O (sd) O(k2 )2 O(a))) .
One particular special case is that of no alternations. Hence, using the
fact that ∃x (P (x) ∨ Q(x)) is equivalent to (∃xP (x)) ∨ (∃xQ(x)), an existential
problem is equivalent to a set 27 of problems of the form
⎛
⎞ ⎛
⎞
∃x ⎝ ∧
f i (x) ≥ 0⎠ ∧ ⎝ ∧
( )
∧
g i (x) = 0⎠ ∧ h i (x) ≠ 0 . (3.65)
f i∈F
g i∈G
h i∈H
This is generally referred to as the existential theory of the reals. Since the
truth of a universal problem is equivalent to the falsity of an existential problem
(∀xP (x) ⇔ ¬∃x¬P (x)), this is all we need to consider.
Given a problem (3.65), cylindrical algebaric decomposition will yield such
an x, if one exists, and failure to yield one is a proof that no such x exists.
However, this is a somewhat unsatisfactory state of affairs in practice, since,
computationally, we are relying not just on the correctness of the theory of
cylindrical algebraic decomposition, but also on the absence of bugs in the implementation.
An alternative is provided by the Positivstellensatz approach [Ste74].
Theorem 26 ([PQR09, Theorem 3]) The set of solutions to (3.65) is empty
if, and only if, there are:
s ∈ con(F ) where con(F ), the cone of F , is the smallest set generated by F and
the set of squares of all elements of R[x] wich is closed under multiplication
and addition;
g ∈ (G) the ideal generated by G;
m ∈ mon(H) where mon(H), the (multiplicative) monoid of H is the set of all
products (inding 1 = the empty product) of elements of H;
such that s + g + m 2 = 0. Furthermore, there is an algorithm to find such s, g
and m (if they exist) in Q[x] provided F , G and H ⊂ Q[x].
27 There may be singly-exponential blow-up here as we convert into disjunctive normal form,
but this is small compared to the other exponential issues in play!