<span class=strong>Tutorial/demonstration session</span>
- 4ti2 – Computing with Lattices, Cones, and Subspaces
Peter Malkin (Université Catholique de Louvain)
In this demonstration, we present the main functionality of the software package 4ti2. Using 4ti2, we can compute Hilbert bases, Graver bases, Groebner bases, and Markov bases of lattices, and the rays of a cone and the circuits of a linear subspace. 4ti2 also has functions to optimize integer linear programs.
- SINGULAR – A Tool for Algebraic Geometry
Hans Schöenemann (Universität Kaiserslautern)
Scenario for SINGULAR presentation
1 Overview on SINGULAR
I will give a short overview of the abilities of Singular,
with special emphasis on the
new features of Singular version 3.0.
The main part of the presentation will concentrate on the following tools for algebraic geometry:
2.1 Non-commutative Extension (G-algebras)
We demonstrate Singular's non-commutative extension
and show an application of it to compute the cohomology of twists of a sheaf
F on Pn associated to coker(M) for M being a graded module.s
It uses the Tate resolution over the exterior algebra.
The normalization will be computed for a reduced ring
The result is a list of rings of the form
The normalization of R/I is the product of the
factor rings of the rings in the list divided out by the ideals.
2.3 Primary Decomposition
There are two algorithms implemented in Singular
which provide primary decomposition:
primdecGTZ, based on Gianni/Trager/Zacharias (written by Gerhard Pfister) and primdecSY, based on Shimoyama/Yokoyama (written by Wolfram Decker and Hans Schoenemann).
We will demonstrate some examples of both.
2.4 Absolute Factorization And Absolute
The absolute factorization of a multivariate polynomial
coefficients in a field K of characteristic zero computes a suitable
field extension L (of K) and computes an absolutely irreducible factor of f in that extension.
These routines are used to compute an absolute primary decomposition:
the routines return an list of he absolute prime components together
with its number of conjugates.
- Demonstration of HomLab
Charles Wampler (General Motors Corporation)
HomLab, a suite of Matlab routines, was created for learning about the numerical solution of polynomial systems using homotopy continuation.The
package is distributed for use with the book Sommese and Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, 2005. In addition to the main software package, the HomLab distribution includes codes to be used in working the exercises at the end of each chapter.
While created for use with the book, HomLab is a general-purpose solver, fast enough for moderately-sized systems. It provides the capability to find all isolated solutions and to provide witness points on positive-dimensional solution sets. Parameter continuation is available for tracking nonsingular isolated solutions through a parameterized family of polynomials, eliminating the overhead that otherwise would be imposed by degenerate and positive-dimensional components.
The software can be downloaded at http://www.nd.edu/~cwample1/HomLab/main.html
The HomLab User's Manual appears as Appendix C of Sommese and Wampler.
- Solving Large Overdetermined Systems with CRACK
Thomas Wolf (Brock University)
Starting as an automatic tools to integrate overdetermined
systems of linear PDEs, CRACK developed in recent years to a package
that is also well suited for polynomially non-linear systems.
Newer algorithms, like one for reducing the number of terms are
essentially different from the Groebner basis algorithm and thus a
useful addition which make the package especially suited for solving
large and heavily overdetermined systems. In the talk and demo an
overview shall be given, featuring, for example, the recent extension
that allows to dynamically generate the system to be solved during its
solution process. This became necessary for a computation in discrete
differential geometry where the polynomial consistency conditions for a
3-dimensional face formula involve 1017 terms.
- Software for Partial Differential Equations
Gregory Reid (University of Western Ontario)
In this talk symbolic software will be described for overdetermined systems of Partial Differential Equations.
The symbolic software is the rifsimp package which is available in distributed
Maple (since Maple 7). It is tightly integrated and automatically called in many applications of Maple (e.g. its ODE and PDE solving routines). It was programmed by Allan Wittkopf.
Examples and applications of the use of this software will be given. Some problems will also be used so that participants can try them during the session.
Examples include: determination of initial data uniqueley determining solutions
of PDE; determination of normal forms for PDE; determination of formal power series for PDE; determination of symmetries of PDE; integration of PDE.
Strategies for tackling difficult problems and examples of their use.
- Computing Tropical Varieties in Gfan
Anders Jensen (Aarhus University)
We will demonstrate how the Gfan software can be used for computing Groebner
fans and tropical varieties of polynomial ideals. For Groebner fans we will see
how to do computations up to symmetry, draw the fans and do interactive Groebner
walks. Tropical computations are more involved. The following problems can be
handled by Gfan: intersecting tropical hypersurfaces, computing tropical bases
of curves and traversing tropical varieties of prime ideals. These will all be
demonstrated. We will discuss the most common problems one encounters with the
tropical part of the software: finding a starting cone, specifying symmetries,
selecting an appropriate output format and reading the output.