Reality and Control of Linear Systems	May 1997, MSRI

-Frank Sottile


During the period of January - May 1997, I examined literally thousands of
cases of the following problem:

Given a 6x2-matrix of integral polynomials F(s), whose maximal minors
have degree 8, and a degree 8 polynomial phi(s) all of whose (distinct)
roots are integers, which 6x4-matrices Coords of local coordinates on
the Grassmannian of 4-planes in C^6 satisfy the following relation:
  
           phi(s) := determinant  [F(s) : Coords].

Specifically, I wanted to know how many of the 14 solutions were real
and how many were complex.

Questions like this arise in the theory of control of linear systems,
and I wanted to use methods at my disposal to investigate these. 
The file poles.html contains a detailed summary of that search.
Briefly, I randomly generated (in SINGULAR) 500 different matrices F(s)
using several different schemes for generating the coefficients.   Then, 
for 400, I considered 25 different 8-tuples of integers as the roots of 
phi(s) (for 100, I considered 50 different 8-tuples).  The file 
compilation.mapl (a maple script) contains the data from that search, 
at least the number of real solutions to each of the 15,000 problems 
considered. 

During this search, I noticed several matrices F(s), which gave the same
number of real points for each polynomial phi(s) tested.  All of these were
examined further for many other polynomials phi(s), and for each, there were
polynomials phi(s) that gave different numbers of real and complex roots.  
In all, there were about 8,000 further evaluations.

This leads to the following question:

What is the behaviour of the number of real vs. complex roots
for a given real matrix of polynomials F(s)?

From Shapiros's conjecture, if F(s) osculates the rational normal curve,
then we always expect the same, maximal, number of real roots.

Are there any other F(s) which always give the same number of real
roots, or do other F(s) always give different numbers of real roots, for
different phi(s)?

Perhaps the cleanest possible answer is that there is a dichotomy:

Either a matrix F(s) comes from planes osculating the rational normal
curve, and hence always gives all real solutions, 

or

Every possible number of real solutions occurs for F(s) and different
phi(s).


I have also investigated Shapiros's conjecture in a number of cases.
That search is detailed elsewhere.