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The Hypersurface conjecture of Shapiro and Shapiro Previous Next Contents

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, ..., Kmp are chosen in a particular fashion, then all dm,p   p-planes H which meet each Ki 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, ..., smp are general distinct real numbers, then each of the dm,p solutions to the system of polynomials
f(s1) = f(s2) = ... = f(smp) = 0
is real.
Equivalently, every p-plane H which meets each K(si) 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.


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