
origami equations
The link above leads to a list of systems of polynomial equations
for the twofolds described in the article "One and TwoFold Origami Axioms"
by R. C. Alperin and R. J. Lang which appears
in the 2006 4OSME Proceedings, Pasadena, CA, A.K.Peters, 2007.
Thanks to Bernd Sturmfels for forwarding this nice family of systems.
Bernd wonders who can solve these equations fastest and most reliably.
Does the number of complex roots always match the Bernstein bound?
What about the number of real roots?
Everyone who has some results on these origami equations,
please do communicate directly with Roger Alperin.
(alperin@math.sjsu.edu)

a problem in Schubert calculus
Thanks to Michael E. O'Sullivan (mosulliv@sciences.sdsu.edu)
for posing this problem.

solve systems of algebraic equations with
parameters obtained by discretizing nonlinear partial differential
equations.
Study how solutions depend on parameters (local and global
bifurcation problems).
Thanks to Nobuki Takayama(takayama@math.kobeu.ac.jp)
for posing this problem.

systems of Marshall Hampton
Clicking on the link above leads to a Mathematica notebook which generates
various polynomial systems. Doing unzip on the file
GeneralNewton3N.zip will also
generate the polynomial systems for you.
An addendum contains a reference to
a paper and another system.
Thanks to Marshall Hampton
(mhampton@d.umn.edu) for contributing these polynomial systems.

applications that lead to overdetermined bilinear algebraic systems
The systems
cover a wide rang of difficulty, from systems with less than 10 unknowns
to systems with over 450 unknowns and nearly 1000 polynomial equations.
They are
all solved in full generality, many systems having no solutions, many having
some and many having a lot of solutions. The systems and their solutions are
in machine readable form and could be used for comparisons between different
software.
The applications concern the classifications of integrable systems. In all
these classifications a polynomial ansatz for the equation or system of
equations was made as well as for a higher order symmetry or a polynomial
first integral.
Another problem concerns the
computation of all conservation laws.
Next is a system of 15 first order PDEs.
The following is an overdetermined algebraic system
1358 polynomial equations for 262 unknowns in an attempt to
find the Cartan Weyl basis of a Lie algebra of 14 Killing vectors
in a certain 8dimensional metric (in collaboration with Dmitry
Galtsov/Moscow).
In an automatic run CRACK can solve many cases but does
currently (Oct 2006) not get through the whole computation.
Thanks to Thomas Wolf (twolf@brocku.ca)
for contributing these problems.
 Systems from Voronoi diagriams
The first problem concerns real solving a univariate polynomial (actually 2)
and the second real solving trivariate polynomial systems, that occur
during the computation of the Voronoi diagram of ellipses in the plane.
An additional system is given by
gzipped Maple code.
Thanks to Elias Tsigaridas
(Elias.Tsigaridas@sophia.inria.fr)
for these probems.

a problem with rational functions
This problem comes from Nicola Sottocornola and may be described in
several ways.
Thanks to Dave Rusin (rusin@ima.umn.edu) for this problem.
 a problem about nilpotent matrices
Thanks to Kenneth Driessel (driessel@ima.umn.edu)
for contributing this problem.
This document is maintained by Jan Verschelde
(jan@math.uic.edu).