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3.5. EQUATIONS AND INEQUALITIES 101
Hence we can essentially introduce the constraint x ≥ 0 by adding a new
variable y and the equation y 2 − x = 0. We can also introduce the constraint
x ≠ 0 by adding a new variable z and xz − 1 = 0 (essentially insisting that x
be invertible). Hence x > 0 can be introduced. Having seen that ≥ and > can
creep in through the back door, we might as well admit them properly, and deal
with the language of real closed fields, i.e. the language of fields (definition 13)
augmented with the binary predicate > and the additional laws:
1. Precisely one of a = b, a > b and b > a holds;
2. a > b and b > c imply a > c;
3. a > b implies a + c > b + c;
4. a > b and c > 0 imply ac > bc.
This is the domain of real algebraic geometry, a lesser-known, but very important,
variant of classical algebraic geometry. Suitable texts on the subject
are [BPR06, BCR98]. However, we will reserve the word ‘algebraic’ to mean
a set defined by equalities only, and reserve semi-algebraic for the case when
inequalities (or inequations 18 ) are in use. More formally:
Definition 65 An algebraic proposition is one built up from expressions of the
form p i (x 1 , . . . , x n ) = 0, where the p i are polynomials with integer coefficients,
by the logical connectives ¬ (not), ∧ (and) and ∨ (or). A semi-algebraic proposition
is the same, except that the building blocks are expressions of the form
p i (x 1 , . . . , x n )σ0 where σ is one of =, ≠, >, ≥, 0 has to be translated as (q > 0 ∧ p >
0) ∨ (q < 0 ∧ p < 0), which is not true when q = 0. If this is not what we mean,
e.g. when p and q have a common factor, we need to say so.
3.5.1 Applications
It runs out that many of the problems one wishes to apply computer algebra to
can be expressed in terms of real semi-algebraic geometry. This is not totally
surprising, since after all, the “real world” is largely real in the sense of R.
Furthermore, even if problems are posed purely in terms of equations, there
may well be implicit inequalities as well. For example, it may be implicit that
18 Everyone agrees that an equation a = b is an equality. a > b and its variants are traditionally
referred to as inequalities. This only leaves the less familiar inequation for a ≠ b.
Some treatments ignore inequations, since a ≠ b = a > b ∨ a < b, but in practice it is useful
to regard inequations as first-class objects.