1.4. THE WRONSKI MAP AND THE SHAPIRO CONJECTURE 5 Theorem 1.5. X(l, n) e2+3 4 2 (l) 2 nl. 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)!
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