Recall from Section 4.ii that the
kplanes in C^{n} meeting
k(nk) general (nk)planes nontrivially
is a complementary dimensional linear section of the Grassmannian,
the intersection of Gr(k,n) with a linear subspace
M of codimension k(nk).
The number of such kplanes is the degree d(k,n)
of the Grassmannian Gr(k,n).
This is also the degree of the linear projection
An important such linear projection is the Wronski map.
Let L(t) be the (nk)plane osculating the
rational normal curve (5.1).
By (4.6) and the discussion
following Theorem 5.2,
the equation for a kplane K to meet L(t)
nontrivially is

(6.1) 
To see why this is called the Wronski map, consider
Gr(k,n) as the set
of kplanes in the space of polynomials of degree at most n1.
Given a kplane
W(K) = det 

Since real manifolds are not necessarily orientable, the degree of a map is a Z/2Zvalued invariant. However, Kronecker [Kr] defined the degree of a regular map RP^{2} > RP^{2} which he called the characteristic, and his definition makes sense for many maps f : X > Y of (not necessarily orientable) compact manifolds.
First suppose that X is oriented.
For a regular value y in Y of f, define
char(f) :=  sgn( det df_{x} ) , 
When X may not be orientable, let oX > X and oY > Y be the canonical spaces of orientations of X and Y, which are 2 to 1 coverings with covering group Z/2Z. These spaces of orientations are canonically oriented. The map f is orientable if it has a lift of : oX > oY that is equivariant with respect to the covering group Z/2Z. Define the characteristic of an orientable map f to be the characteristic of the orienting lift, of. This is welldefined, even if oY consists of 2 components. This characteristic satisfies a fundamental inequality.
We call this characteristic the real degree of the map f
Consider this notion for the Wronski map of the real Grassmannian
Gr_{R}(k,n).
The Grasmannian has dimension
k(nk) and is orientable if and only if n is even.
Eremenko and Gabrielov [EG2]
compute the degree of the Wronski map.
To state their main result, we introduce some additional combinatorics.
Recall the interpretation (4.9)
I(k,n)(t) :=  t^{c} , 
They prove this by an induction reminiscent of that
used in the proof of Theorem 4.6.
In the course of their proof, they construct a polynomial f with all
roots real having d(k,n) (= degree of the Grassmannian)
real points in
q_{W}^{1}(f) (one for each chain in the
Bruhat order) and show that
White [Wh] studied the statistic I(k,n)(1) and showed that it equals zero if and only if n is even. The main result of Eremenko and Gabrielov is the following, which gives a lower bound for the number of real solutions in some cases of the Shapiro conjecture.
This lower bound holds not only if the osculating planes are real, but if the set of points {t_{1}, t_{2}, ..., t_{k(nk)}} where they osculate are real (the roots of a real polynomial), which is equivalent to the consition that this set is stable under complex conjugation.
Consider now a linear projection different than the Wronski map. If we move the center E, then the degree of the projection q_{E} does not change as long as E does not meet the Grassmannian. Thus if E is in the same component of the space of codimension k(nk)+1 linear subspaces not meeting the Grassmannian as is the center W of the Wronski map, then COrollary 6.6 applies to q_{E}^{1}(x), for x a real point in the image of q_{E}.
These new ideas of Eremenko and Gabrielov, particularly their notion of real degree, greatly increase our understanding of the real Schubert calculus. We emphasize that this is an important start in the search for lower bounds to other enumerative problems. Not all enumerative problems involve a linear projection of a submanifold of projective space. In fact, of the other enumerative problems we have considered in the Schubert calculus, only those simple Schubert calculus problems on Lagrangian Grassmannians involve linear projections, and they have a lower bound of 0. There has been very little experimental work investigating lower bounds. We describe some of it in the final section.