Contents - Student subdomain for University of Bath

3.5. EQUATIONS AND INEQUALITIES 105
The reader may complain that example 3 is overly complex: can’t we just write
{f > 0 ∧ x < −2} ∪ {x = − √ 3 + √ 2} ∪ {f < 0 ∧ x < 0} ∪
{x = − √ 3 − √ 2 < 0} ∪ {f > 0 ∧ x > −2 ∧ x < 2} ∪
{x = √ 3 − √ 2} ∪ {f < 0 ∧ x > 0} ∪ {x = √ 3 + √ 2} ∪ {f > 0 ∧ x > 2}?
In this case we could, but in general theorem 8 means that we cannot 21 : we need
RootOf constructs, and the question then is “which root of . . .”. In example
3, we chose to use numeric inequalities (and we were lucky that they could be
chosen with integer end-points). It is also possible [CR88] to describe the roots
in terms of the signs of the derivatives of f, i.e.
{f > 0 ∧ x < −2} ∪ {f = 0 ∧ f ′ < 0 ∧ f ′′′ < 0} ∪ {f < 0 ∧ x < 0} ∪
{f = 0 ∧ f ′ > 0 ∧ f ′′′ < 0} ∪ {f > 0 ∧ x > −2 ∧ x < 2} ∪
{f = 0 ∧ f ′ < 0 ∧ f ′′′ > 0} ∪ {f < 0 ∧ x > 0} ∪
{f = 0 ∧ f ′ > 0 ∧ f ′′′ > 0} ∪ {f > 0 ∧ x > 2}
(as it happens, the sign of f ′′ is irrelevant here). This methodology can also
be applied to the one-dimensional regions, e.g. the first can also be defined as
{f > 0 ∧ f ′ > 0 ∧ f ′′ < 0 ∧ f ′′′ < 0}.
We may ask how we know that we have a decomposition, and where these
extra constraints (such as x > 0 in example 5 or x < 3 2
in example 6) come
from. This will be addressed in the next section, but the brief answers are:
• we know something is a decomposition because we have constructed it
that way;
• x = 0 came from the leading coefficient (with respect to y) of xy − 1,
whereas 3 2 in example 6 is a root of Disc y(f).
We stated in definition 66 that the cells must be non-empty. How do we
know this? For the zero-dimensional cells {f = 0 ∧ x > a ∧ x < b}, we can rely
on the fact that if f changes sign between a and b, there must be at least one
zero, and if f ′ does not 22 , there cannot be more than one: such an interval can
be called an isolating interval. In general, we are interested in the following
concept.
Definition 67 A sampled algebraic decomposition of R n is an algebraic decomposition
together with, for each cell C, an explicit point Sample(C) in that
cell.
By ‘explicit point’ we mean a point each of whose coordinates is either a rational
number, or a precise algebraic number: i.e. a defining polynomial 23 together
with an indication of which root is meant, an isolating interval, a sufficiently
21 And equation (3.11) means that we probably wouldn’t want to even when we could!
22 Which will involve looking at f ′′ and so on.
23 Not necessarily irreducible, though it is normal to insist that it be square-free.