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,R m+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 Wr−1(Φ), R 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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