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Software for Algebraic Geometry
23-27 Oct 2006

1. origami equations
The link above leads to a list of systems of polynomial equations for the two-folds described in the article "One and Two-Fold 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)

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

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

4. 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.

5. applications that lead to overdetermined bi-linear 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 8-dimensional 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.

6. 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.

7. 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.

8. 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).
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