1.5. LOWER BOUNDS 11
The idea behind the proof of Theorem 1.14 is to compute the topological de-
gree of the real Wronski map, which is the restriction of the Wronski map to real
subspaces of polynomials,
WrR := Wr|Gr(m,Rm+p−1[t]) : Gr(m, Rm+p−1[t]) −→
RPmp
.
This maps the Grassmannian of real subspaces of polynomials of degree m+P −1
to the space of real polynomials of degree mp. Recall that the topological degree
(or mapping degree) of a map f : X Y between two oriented manifolds X and
Y of the same dimension is the number d such that f∗[X] = d[Y ], where [X] and
[Y ] are the fundamental cycles of X and Y in homology, respectively, and f∗ is the
functorial map in homology. When f is differentiable, this mapping degree may be
computed as follows. Let y Y be a regular value of f so that the derivative map
on tangent spaces dfx : TxX TyY is an isomorphism at any point x in the fiber
f
−1(y)
above y. Since X and Y are oriented, the isomorphism dfx either preserves
the orientation of the tangent spaces or it reverses the orientation. Let P be the
number of points x f
−1(y)
at which dfx preserves the orientation and R be the
number of points where the orientation is reversed. Then the mapping degree of f
is the difference P R.
There is a slight problem in computing the mapping degree of WrR, as neither
the real Grassmannian GrR nor the real projective space RPmp are orientable when
m+p is odd, and thus the mapping degree of WrR is not defined when m+p is odd.
Eremenko and Gabrielov get around this by computing the degree of the restriction
of the Wronski map to open cells of GrR and RPmp, where WrR is a proper map.
They also show that it is the degree of a lift of the Wronski map to oriented double
covers of both spaces. The degree bears a resemblance to the Welschinger invariant
as it has the form |

±1|, the sum over all real points in WrR
−1(Φ),
for Φ a regular
value of the Wronski map. This resemblance is no accident. Solomon [136] showed
how to orient a moduli space of rational curves with marked points so that the
Welschinger invariant is indeed the degree of a map.
While both of these examples of geometric problems possessing a lower bound
on their numbers of real solutions are quite interesting, they are rather special. The
existence of lower bounds for more general geometric problems or for more general
systems of polynomials would be quite important in applications, as these lower
bounds guarantee the existence of real solutions.
With Soprunova, we [138] set out to develop a theory of lower bounds for sparse
polynomial systems, using the approach of Eremenko and Gabrielov via mapping
degree. This is a first step toward practical applications of these ideas. Chapters 7
and 8 will elaborate this theory. Here is an outline:
(i) Realize the solutions to a system of polynomials as the fibers of a map
from a toric variety.
(ii) Characterize when a toric variety (or its double cover) is orientable, thus
determining when the degree of this map (or a lift to double covers) exists.
(iii) Develop a method to compute the degree in some (admittedly special)
cases.
(iv) Give a nice family of examples to which this theory applies.
(v) Use the sagbi degeneration of a Grassmannian to a toric variety [154,
Ch. 11] and the systems of (iv) to reprove the result of Eremenko and
Gabrielov.
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