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106 CHAPTER 3. POLYNOMIAL EQUATIONS
exact 24 numerical approximation or a Thom’s Lemma [CR88] list of signs of
derivatives.
Definition 68 A decomposition D of R n is said to the sign-invariant for a
polynomial p(x 1 , . . . , x n ) if and if only if, for each cell C ∈ D, precisely one of
the following is true:
1. ∀x ∈ C p(x) > 0;
2. ∀x ∈ C p(x) < 0;
3. ∀x ∈ C p(x) = 0;
It is sign-invariant for a set of polynomials if, and only if, for each polynomial,
one of the above conditions is true for each cell.
It therefore follows that, for a sampled decomposition, the sign throughout the
cell is that at the sample point.
3.5.4 Cylindrical Algebraic Decomposition
Notation 18 Let n > m be positive natural numbers, and let R n have coordinates
x 1 , . . . , x n , with R m having coordinates x 1 , . . . , x m .
Definition 69 An algebraic decomposition D of R n is said to be cylindrical
over a decomposition D ′ of R m if the projection onto R m of every cell of D is a
cell of D ′ . The cells of D which project to C ∈ D ′ are said to form the cylinder
over C, denoted Cyl(C). For a sampled algebraic decomosition, we also insist
that the sample point in C be the projection of the sample points of all the cells
in the cylinder over C.
Cylindricity is by no means trivial.
Example 4 Consider the decomposition of R 2 = S 1 ∪ S 2 ∪ S 3 where
S 1 = {(x, y) | x 2 + y 2 − 1 > 0},
S 2 = {(x, y) | x 2 + y 2 − 1 < 0},
S 3 = {(x, y) | x 2 + y 2 − 1 = 0}.
This is an algebraic decomposition, and is sign-invariant for x 2 + y 2 − 1. However,
it is not cylindrical over any decomposition of the x-axis R 1 . The projection
of S 2 is (−1, 1), so we need to decompose R 1 as
(−∞, −1) ∪ {−1} ∪ (−1, 1) ∪ {1} ∪ (1, ∞). (3.58)
24 By this, we mean an approximation such that the root cannot be confused with any other,
which generally means at least an approximation close enough that Newton’s iteration will
converge to the indicated root. Maple’s RootOf supports such a concept.