8 1. OVERVIEW

(1)

(−1)

(−1)

(0)

γ

(s)

Q

γ(s)

✻

Figure 1.2. The fourth tangent line meets hyperboloid in two real points.

defines a map ρ:

P1

→

P1

whose critical points are those t for which ρ (t) = 0. Since

ρ (t) = (f g −g

f)/g2,

we see that the critical points are the roots of the Wronskian

of f and g. Composing the rational function ρ:

P1

→

P1

with an automorphism

of the target

P1

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