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108 CHAPTER 3. POLYNOMIAL EQUATIONS
(where the Q i are ∀ or ∃) is
∨
C ′ ∈D 0∀∃C∈Cyl k (C ′ )P (Sample(C))
Def(C ′ ), (3.64)
where by ∀∃ we mean that we are quantifying across the coordinates of Sample(C)
according to the quantifiers in (3.63).
We can use the (sampled) cylindrical algebraic decomposition in example 4 to
answer various questions.
Example 5 ∀y x 2 +y 2 −1 > 0. For the sampled cells 〈(−∞, −1), (x = −2, y =
0)〉 and 〈(1, ∞), (x = 2, y = 0)〉, the proposition is true at the sample points,
hence true everywhere in the cell. For all the other cells in (3.58), there is a
sample point for which it is false (in fact, y = 0 always works). So the answer
is (−∞, −1) ∪ (1, ∞).
Example 6 ∃y x 2 + y 2 − 1 > 0. For every cell in (3.58), there is a sample
point above it for which the proposition is true, hence we deduce that the answer
is (3.58), which can be simplified to true.
We should note (and this is both one of the strengths and weaknesses of this
approach) that the same cylindrical algebraic decomposition can be used to
answer all questions of this form with the same order of (blocks of) quantified
variables, irrespective of what the quantifiers actually are.
Example 7 (∃y x 2 + y 2 − 1 > 0) ∧ (∃y x 2 + y 2 − 1 < 0). This formula
is not directly amenable to( this approach, since it is not in prenex form. In
prenex form, it is ∃y 1 ∃y 2 (x 2 + y1 2 − 1 > 0) ∧ (x 2 + y2 2 < 0) ) and we need an
analogous 25 decomposition of R 3 cylindric over R 1 . Fortunately, (3.58) suffices
for our decomposition of R 1 , and the answer is (−1, 1), shown by the sample
point (x = 0, y 1 = 2, y 2 = 0), and by the fact that at other sample points of R 1 ,
we do not have y 1 , y 2 satisfying the conditions.
However, it could be argued that all we have done is reduce problem 3 to
the following one.
Problem 4 Given a quantified semi-algebraic proposition as in theorem 25,
produce a sign-invariant decomposition D k cylindrical over the appropriate D i
such that theorem 25 is applicable. Furthermore, since theorem 25 only talks
about “a” quantifier-free form, we would like the simplest possible such D k (see
[Laz88]).
we will call it block-
There is no common term for such a decomposition:
cylindrical.