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
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