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102 CHAPTER 3. POLYNOMIAL EQUATIONS
quantities are non-negative, or that concentrations is biochemistry lie in the
range [0, 1].
Robot motion planning . . .
It is also often important to prove unsatisfiability, i.e. that a semi-algebraic
formula has no solutions. [Mon09] gives several examples, ranging from program
proving to biological systems. The program proving one is as follows. One
wishes to prove that I is an invariant (i.e. if it was true at the start, it is true
at the end) of a program which moves from one state to another by a transition
relation τ. More formally, one wishes to prove that there do not exist two states
s, s ′ such that s ∈ I, s ′ /∈ I, but s → τ s ′ . Such a pair (s, s ′ ) would be where “the
program breaks down”, so a proof of unsatisfiability becomes a proof of program
correctness. This places stress on the concept of ‘proof’ — “I can prove that
there are no bad cases” is much better than “I couldn’t find any bad cases”.
3.5.2 Quantifier Elimination
A fundamental result of algebraic geometry is the following, which follows from
the existence of resultants (section A.1).
Theorem 24 A projection of an algebraic set is itself an algebraic set.
For example, the projection of the set defined by
{
}
(x − 1) 2 + (y − 1) 2 + (z − 1) 2 − 4, x 2 + y 2 + z 2 − 4
(3.48)
on the x, y-plane is the ellipse
8 x 2 + 8 y 2 − 7 − 12 x + 8 xy − 12 y. (3.49)
We can regard equation (3.48) as defining the set
(
)
∃z (x − 1) 2 + (y − 1) 2 + (z − 1) 2 = 4 ∧ x 2 + y 2 + z 2 = 4
(3.50)
and equation (3.49) as the quantifier-free equivalent
8 x 2 + 8 y 2 − 12 x + 8 xy − 12 y = 7. (3.51)
Is the same true in real algebraic geometry? If P is a projection operator,
and R denotes the real part, then clearly
P (R(U) ∩ R(V )) ⊆ R(P (U ∩ V )). (3.52)
However, the following example shows that the inclusion can be strict. Consider
{
}
(x − 3) 2 + (y − 1) 2 + z 2 − 1, x 2 + y 2 + z 2 − 1
Its projection is (10 − 6 x − 2 y) 2 , i.e. a straight line (with multiplicity 2). If we
substitute in the equation for y in terms of x, we get z = √ −10 x 2 + 30 x − 24,