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The conjecture of Shapiro and Shapiro for general Schubert
conditions
3.ii. The conjecture of Shapiro and Shapiro
The conjecture of Shapiro and Shapiro makes sense in the wider context
of general Schubert conditions.
Given a point s on a real rational normal curve, we can consider the
flag of subspaces F.(s) osculating the curve at
s.
A practical way to generate such flag of subspaces is to begin with a
parameterization of a real rational normal curve as a row vector,
and then let Fi(s) be the linear span of this
vector and its first i-1 derivatives with respect to the parameter.
Conjecture 2. (Shapiro-Shapiro)
Suppose that
a1, a2,
..., an
are Schubert data
(|a1| + |a2|
+ ... + |an| =
mp)
and s1, s2, ...,
sn
are general distinct real numbers, then each of the finitely many
p-planes H satisfying the Schubert conditions
ai with respect to the flags
F.(si)
are real.
Equivalently, the common intersection of the Schubert varieties
|
Ya1F.1,
Ya2F.2,
...,
YanF.n,
|
consists solely of real p-planes H.