Acknowledgements
I thank Boris Shapiro for sharing these conjectures
and Joachim Rosenthal with whom I found the first strong evidence for their
validity [RS].
The amazing computation of Faugère, Rouillier, and Zimmermann
[FRZ]
inspired me to describe my own explorations on this conjecture.
Bernd Sturmfels helped me learn computer algebra and
suggested using discriminants.
This work also benefitted from discussions with
David Eisenbud, Ioannis Emiris, Jean-Charles Faugère, Bill
Fulton, Birk Huber, Bernard Mourrain, Bruce Reznick,
Fabrice Rouillier,
Jan Verschelde,
and Paul Zimmermann.
Lastly, this would not be possible were it not
for the authors of the following software packages:
Linux, Maple,
SINGULAR,
Macaulay2,
MuPAD, FGB, RealSolving,
PHC,
and also my brothers Larry Sottile and Joe Sottile who helped me build the
computer which performed most of these experiments.
I gratefully acknowledge the support of
MSRI, where this project began and
was completed, and the NSF under grants DMS-9022140 and DMS-9701755,
and also the mathematics department of the
University of Toronto
and NSERC of Canada under grant number OGP0170279.