The Hypersurface conjecture of Shapiro and Shapiro

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2.ii. The conjecture of Shapiro and Shapiro

Boris Shapiro and Michael Shapiro made a precise conjecture which asserts that if the m-planes K₁, ..., K_mp are chosen in a particular fashion, then all d_m,p p-planes H which meet each K_i nontrivially are real.

Let K(s) be the following m by (m+p)-matrix of polynomials: (Also the row space of the same matrix, a m-plane.)

Let f(s) be the polynomial

Conjecture 1. (Shapiro-Shapiro)
If s₁, s₂, ..., s_mp are general distinct real numbers, then each of the d_m,p solutions to the system of polynomials

f(s₁) = f(s₂) = ... = f(s_mp) = 0
is real.
Equivalently, every p-plane H which meets each K(s_i) nontrivially is real.

This is true when (m,p)=(2,2). We have a Maple V.5 script, which, when run, proves this assertion. Here is its output.