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3.5. EQUATIONS AND INEQUALITIES 103
which is never real for real x. In fact R(U) ∩ R(V ) = ∅, as is obvious from
the geometric interpretation of two spheres of radius 1 centred at (0, 0, 0) and
(3, 1, 0). Hence the methods we used for (complex) algebraic geometry will not
translate to real algebraic geometry.
The example of y 2 − x, whose projection is x ≥ 0, shows that the projection
of an algebraic set need not be an algebraic set, but might be a semi-algebraic
set. Is even this guaranteed? What about the projection of a semi-algebraic
set? In the language of quantified propositions, we are asking whether, when F
is an algebraic or semi-algebraic proposition, the proposition
∃y 1 . . . ∃y m F (y 1 , . . . , y m , x 1 , . . . , x n ) (3.53)
has a quantifier-free equivalent G(x 1 , . . . , x n ), where G is a semi-algebraic proposition.
We can generalise this.
Problem 3 (Quantifier Elimination) Given a quantified proposition 19
Q 1 y 1 . . . Q m y m F (y 1 , . . . , y m , x 1 , . . . , x n ), (3.54)
where F is a semi-algebraic proposition and the Q i are each either ∃ or ∀, does
there exist a quantifier-free equivalent semi-algebaric proposition G(x 1 , . . . , x n )?
If so, can we compute it?
The fact that there is a quantifier-free equivalent is known as the Tarski–
Seidenberg Principle [Sei54, Tar51]. The first constructive answer to the question
was given by Tarski [Tar51], but the complexity of his solution was indescribable
20 . A better (but nevertheless doubly exponential) solution had
to await the concept of cylindrical algebraic decomposition (CAD) [Col75] described
in the next section.
Notation 17 Since ∃x∃y is equivalent to ∃y∃x, and similarly for ∀, we extend
∃ and ∀ to operate on blocks of variables, so that, if x = (x 1 , . . . , x n ), ∃x is
equivalent to ∃x 1 . . . ∃x n . If we use this notation to rewrite equation 3.54 with
the fewest number of quantifiers, the quantifiers then have to alternate, so the
formula is (where the y i are sets of variables)
∀y 1 ∃y 2 ∀y 3 . . . F (y 1 , y 2 , . . . , x 1 , . . . , x n ), (3.55)
or
∃y 1 ∀y 2 ∃y 3 . . . F (y 1 , y 2 , . . . , x 1 , . . . , x n ). (3.56)
In either form, the number of quantifiers is one more than the number of alternations.
19 Any proposition with quantified variables can be converted into one in this form, so-called
prenex normal form — see any standard logic text.
20 In the formal sense, that there was no elementary function which could describe it, i.e.
no tower of exponentials of fixed height would suffice!