Theorem 1.5. X(l, n)
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
That is, the differences of vectors in A generate
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
admits a regular trian-
gulation with each simplex having minimal volume
. Then there is a system of
sparse polynomials with support Δ
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
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
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])
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)!
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