The full Shapiro Conjecture concerns
p-planes satisfying general incidence conditions
imposed by linear subspaces osculating the real rational normal
curve.
It asserts that when there are finitely many such p-planes,
they will all be real.
There is amazing experimental evidence.
In the 1990's a few million cases of the conjecture for
general incidence conditions
were tested, and all supported the conjecture.
More recently, 132.919.238 instances of the
conjecture for general incidence conditions
were tested on 212 different geometric problems on the
Grassmannians of 3-planes in 6- and 7-space, and 4-planes in 8-space.
This took 3.57 gigaHertz-years of CPU time.
It is also true asymptotically:
Every p-plane meeting mp planes osculating the real
rational normal curve is real,
if the points of osculation are sufficiently close together.
Eremenko and Gabrielov's result on rational functions
proves it when
either of p or m is equal to 2.
