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)
ands1, 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
