Figure 1.2. The fourth tangent line meets hyperboloid in two real points.
defines a map ρ:

whose critical points are those t for which ρ (t) = 0. Since
ρ (t) = (f g −g
we see that the critical points are the roots of the Wronskian
of f and g. Composing the rational function ρ:

with an automorphism
of the target
gives an equivalent rational function, and the equivalence class of
ρ is determined by the linear span of its numerator and denominator. Thus the
Shapiro Conjecture asserts that a rational function having only real critical points
is equivalent to a real rational function.
Eremenko and Gabrielov [46] proved exactly this statement in 2002, thereby
establishing the Shapiro Conjecture in the case m = 2.
Theorem 1.12. A rational function with only real critical points is equivalent
to a real rational function.
In Chapter 11 we will present an elementary proof of this result that Eremenko
and Gabrielov gave in 2005 [42].
1.5. Lower bounds
We begin with perhaps the most exciting recent development in real algebraic
geometry. This starts with the fundamental observation of Euclid that two points
determine a line. Slightly less elementary is that five points in the plane with no
three collinear determine a conic. In general, if you have n general points in the
plane and you want to pass a rational curve of degree d through all of them, there
may be no solution to this interpolation problem (if n is too big), or an infinite
number of solutions (if n is too small), or a finite number of solutions (if n is just
right). It turns out that “n just right” means n = 3d−1 (n = 2 for lines where
d = 1, and n = 5 for conics where d = 2).
A harder question is, if n = 3d−1, how many rational curves of degree d
interpolate the points? Call this number Nd, so that N1 = 1 and N2 = 1 because
the line and conic of the previous paragraph are unique. It has long been known
that N3 = 12 (see Example 9.3 for a proof), and in 1873 Zeuthen [164] showed
that N4 = 620. That was where matters stood until 1989, when Ran [118] gave
a recursion for these numbers. In the 1990’s, Kontsevich and Manin [88] used
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