Enumerative Real Algebraic Geometry: Rational functions with real critical points.

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.

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 P^d 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 P^d meeting 2d-2 general codimension 2 planes. The answer is the dth Catalan number, C_d, given in (4.1).

Consider the above description in the affine cone over P^d . 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)) = Y_a 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 C_d 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 C_d 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 P^m and the composition is a parameterized rational curve in P^m. Post-composition by an element of Aut(P^m) 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 P^m. This connection between linear series and the Schubert calculus originated in work of Castelnuovo [Ca].

5.ii. Rational functions with real critical points.