1.4. THE WRONSKI MAP AND THE SHAPIRO CONJECTURE 5

Theorem 1.5. X(l, n)

e2+3

4

2(2)nl.l

For small values of l, it is not hard to improve this. For example, when l = 0,

the support A of the system is a simplex, and there will be at most one positive

real solution, so X(0,n) = 1. Theorem 1.5 was inspired by the sharp bound of

Theorem 1.7 when l = 1 [15]. A set A of exponents is primitive if A aﬃnely spans

the full integer lattice

Zn.

That is, the differences of vectors in A generate

Zn.

Theorem 1.6. If l = 1 and the set A of exponents is primitive, then there can

be at most 2n+1 nondegenerate nonzero real solutions, and this is sharp in that

for any n there exist systems with n+2 monomials and 2n+1 nondegenerate real

solutions whose exponent vectors aﬃnely span Zn.

Observe that this bound is for all nonzero real solutions, not just positive

solutions. We will discuss this in Section 5.3. Further analysis by Bihan gives the

sharp bound on X(1,n).

Theorem 1.7 (Bihan [15]). X(1,n) = n + 1.

These fewnomial bounds are discussed and proven in Chapters 5 and 6.

In contrast to these results establishing absolute upper bounds on the number

of real solutions which improve the trivial bound of the number d of complex roots,

there are a surprising number of problems that come from geometry for which all

solutions can be real. For example, Sturmfels [153] proved the following. (Regular

triangulations are defined in Section 4.2, and we give his proof in Section 4.4.)

Theorem 1.8. Suppose that a lattice polytope Δ ⊂

Zn

admits a regular trian-

gulation with each simplex having minimal volume

1

n!

. Then there is a system of

sparse polynomials with support Δ ∩

Zn

having all solutions real.

For many problems from enumerative geometry, it is similarly possible that all

solutions can be real. This will be discussed in Chapter 9.

1.4. The Wronski map and the Shapiro Conjecture

The Wronskian of univariate polynomials f1(t),...,fm(t) is the determinant

Wr(f1,f2,...,fm) := det

(

(

d

dt

)i−1fj

(t)

)

i,j=1,...,m

.

When the polynomials fi have degree m+p−1 and are linearly independent, the

Wronskian has degree at most mp. For example, if m = 2, then Wr(f, g) = f g−fg ,

which has degree 2p as the coeﬃcients of

t2p+1

in this expression cancel. Up to

a scalar, the Wronskian depends only upon the linear span of the polynomials

f1,...,fm. Removing these ambiguities gives the Wronski map,

(1.4) Wr : Gr(m, Cm+p−1[t]) −→ P(Cmp[t])

Pmp

,

where Gr(m, Cm+p−1[t]) is the Grassmannian of m-dimensional subspaces of the

linear space Cm+p−1[t] of complex polynomials of degree m+p−1 in the variable

t, and P(Cmp[t]) is the projective space of complex polynomials of degree at most

mp, which has dimension mp, equal to the dimension of the Grassmannian.

Work of Schubert in 1886 [130], combined with a result of Eisenbud and Harris

in 1983 [40] shows that the Wronski map is surjective and the general polynomial

Φ ∈ Pmp has

(1.5) #m,p :=

1!2! · · · (m−1)! · (mp)!

m!(m+1)! · · · (m+p−1)!