Shapiro Conjecture : Secant Lines

Secant Lines

This last example was generalized by Eremenko, Gabrielov, M. Shapiro, and Vainstein.

A collection of 2n-2 lines m₁, m₂, ..., m_2n-2 which are secant to the rational curve is separated if there are 2n-2 disjoint intervals I₁, I₂, ..., I_2n-2 on the rational normal curve g(RP¹) so that line m_i meets g(RP¹) at two points of interval I_i.

Theorem (Eremenko, Gabrielov, Shapiro, and Vainstein)
Given 2n-2 secant lines to the rational normal curve that are separated, each of the

#_{n-1, 2} =

(2n-2)!

n! (n-1)!

n-2 planes that meet all of the lines are real.

A tangent line is the limit of a secant line, so this recovers Shapiro's conjecture for n-2 planes.
Letting three secant lines become tangent lines is the geometric problem just discussed.