# <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.**2 Applications**

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*P*associated to coker(M) for M being a graded module.s^{n}

It uses the Tate resolution over the exterior algebra.

The normalization will be computed for a reduced ring*R/I*.

The result is a list of rings of the form*R*._{i}/I_{i}

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**

Primary Decomposition

The absolute factorization of a multivariate polynomial*f*with

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 10^{17}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.