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for Grassmannians
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 (nk) 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 : P^{1} > P^{1}.
Two rational functions f_{1} and f_{2}
are equivalent if f_{1} = A(f_{2}),
where A is a fractional linear transformation of P^{1}.
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
P^{1} > P^{d}
  > P^{1} ,

(5.7) 
where the first map is the rational normal curve
g : [s,t] >
[s^{d}, s^{d1}t, ...
st^{d1}, t^{d}] ,
and the second is a linear projection
[x_{0}, x_{1}, ..., x_{d}]
>
[L_{1}(x), L_{2}(x)] ,
where L_{1} and L_{2} are independent linear
forms.
Let E in P^{d}
be the center of this projection,
the linear subspace where L_{1} = L_{2} = 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 P^{1} 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(P^{1}) 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 2d2 critical points?
Reasoning as above, she reduced this to the problem of determining the number
of codimension 2 planes E in P^{d} meet
2d2 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 2d2
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(d1, d+1) giving a rational function
of degree d with critical point at t in
C (contained in P^{1}) are points
in the simple Schubert variety X(F_{2}(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(P^{1}) 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
nk = 2 and each a=1.
By Theorem 5.2, this implies the full
conjecture when nk = 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 2d2 real numbers.
Let f : P^{1} > P^{1} be a real
rational function of degree d with only real critical points.
Then f^{1}(RP^{1}) is a subset of
CP^{1} which is an embedded graph containing
RP^{1} 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 2d2 distinct real
points, they essentially construct a rational function f of degree
d having these critical points with
f^{1}(RP^{1}) a graph in the class G.
A key point involves degenerate rational maps with
fewer than 2d2 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}.
Postcomposition 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
P^{1}.
Such a linear series is
ramified at a point
t in
P^{1} when the center
E meets the
mplane
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].
Next: 5.iii. Generalizations of Shapiros'
Conjecture
Up: 5. The Conjecture of Shapiro and Shapiro:
Table of Contents
Previous: 5.i. The Conjecture of Shapiro and Shapiro
for Grassmannians