Next: 5.iii. Generalizations of Shapiros' Conjecture
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## 5.ii. Rational functions with real critical points.

By far the strongest evidence for Conjecture 5.1 is that it is true when k or (n-k) is equal to 2.

Theorem 5.6 (Eremenko and Gabrielov [EG1])   Conjecture 5.1 is true when one of k or (n-k) is equal to 2.

This is a consequence of a theorem about rational functions with real critical points. A rational function is an algebraic map f : P1 --> P1. Two rational functions f1 and f2 are equivalent if f1 = A(f2), where A is a fractional linear transformation of P1.

Theorem 5.7 ([EG1])   If all the critical points of a rational function are real, then it is equivalent to a real rational function.

Consider the composition

 P1  ----->  Pd  - - ->  P1 , (5.7)

where the first map is the rational normal curve

g : [s,t]  |---->  [sd, sd-1t, ... std-1, td] ,

and the second is a linear projection

[x0, x1, ..., xd]  ---->  [L1(x), L2(x)] ,

where L1 and L2 are independent linear forms. Let E in Pd be the center of this projection, the linear subspace where L1 = L2 = 0. When E is disjoint from the rational normal curve, this composition defines a rational function of degree d, and all such rational functions occur in this manner. In fact, equivalence classes of rational maps are exactly those maps with the same center of projection. A rational function f has a critical point at t in P1 if the differential df vanishes at t. If we consider the composition (5.7), this implies that the center E meets the line tangent to the rational normal curve g(P1) at g(t).

Goldberg [Gol] asked (and answered) the question: how many equivalence classes of rational functions of degree d have a given set of 2d-2 critical points? Reasoning as above, she reduced this to the problem of determining the number of codimension 2 planes E in Pd meet 2d-2 given tangents to the rational normal curve. Formulating this in the dual projective space, we recover the problem of Section 4.i: Determine the number of lines in Pd meeting 2d-2 general codimension 2 planes. The answer is the dth Catalan number, Cd, given in (4.1).

Consider the above description in the affine cone over Pd . The centers E in Gr(d-1, d+1) giving a rational function of degree d with critical point at t in C (contained in P1) are points in the simple Schubert variety X(F2(t)) = Ya F.(t), where a is the simple Schubert condition of Theorem 5.2 and F.(t) is the flag of subspaces osculating the rational normal curve g(P1) at g(t). An equivalence class of rational functions contains a real rational function when the common center E is real. In this way, Theorem 5.7 implies Conjecture 5.1 when n-k = 2 and each |a|=1. By Theorem 5.2, this implies the full conjecture when n-k = 2. Working in the dual space, we deduce Conjecture 5.1 when k = 2.

Gabrielov and Eremenko prove Theorem 5.7 by showing there exist Cd distinct real rational functions with critical points at a given set of 2d-2 real numbers. Let f : P1 --> P1 be a real rational function of degree d with only real critical points. Then f-1(RP1) is a subset of CP1 which is an embedded graph containing RP1 and also stable under complex conjugation whose vertices have valence 4 and are at the critical points. There are exactly Cd isotopy classes of such graphs. For each isotopy class G and collection of 2d-2 distinct real points, they essentially construct a rational function f of degree d having these critical points with f-1(RP1) a graph in the class G. A key point involves degenerate rational maps with fewer than 2d-2 critical points. It may be interesting to relate this to the degenerations in the proof of Theorem 4.6.

Remark 5.8   If we project the rational normal curve from a center E of codimension m+1, then the image of the projection is Pm and the composition is a parameterized rational curve in Pm.   Post-composition by an element of Aut(Pm) defines an equivalence relation on such maps and equivalence classes are determined by the centers of projection. These centers are codimension m+1 linear series of degree d on P1. Such a linear series is ramified at a point t in P1 when the center E meets the m-plane osculating the rational normal curve. A rational curve/linear series is maximally inflected if the ramification is at real points. We just considered the case d=1 of rational functions with real critical points, and the case d=2 was discussed in Section 3.ii. The existence of maximally inflected curves with simple ramification is a consequence of Theorem 5.3, and Conjecture 5.1 predicts this is a rich class of real curves in Pm. This connection between linear series and the Schubert calculus originated in work of Castelnuovo [Ca].

Next: 5.iii. Generalizations of Shapiros' Conjecture