Shapiro Conjecture : Secant Flags

Secant Flags

General incidence conditions on p-planes H are imposed by (full) flags,
which are sequences F. of linear spaces F. := F₀, F₁, F₂, ..., F_n-1, F_n , where F_i has dimension i and contains F_i-1.

The Shapiro Conjecture for general incidence conditions involves flags of subspaces osculating the rational normal curve.

A flag F. is secant to the rational normal curve if each part F_i is spanned by i+1 real points on the rational normal curve.

A collection of secant flags is separated if there are disjoint intervals, one for each flag, such that a flag is secant at points in the corresponding interval.

Conjecture (Eremenko, Gabrielov, Shapiro, and Vainstein)
Every p-plane satisfying incidence conditions imposed by separated secant flags is real (when there are supposed to be finitely many).

An osculating flag is the limit of a secant flag, so this generalizes the Shapiro Conjecture.