Normaliz 2.5

Winfried Bruns (wbruns@uos.de), Bogdan Ichim (bogdan_ichim@yahoo.com) and Christof Söger (csoeger@uos.de)

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Normaliz is a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones.

Its input data can be specified in terms of

• a system of generators or vertices or
• a system of linear homogeneous Diophantine equations, inequalities and congruences or
• a binomial ideal.

Normaliz computes

• the dual cone of a rational cone (in other words, given generators, Normaliz computes the defining hyperplanes, and vice versa)
• a placing triangulation of a vector configuration (resulting in a triangulation of the cone generated by it)
• a shelling of a vector configuration (resulting in a shelling of the cone generated by it)
• the Hilbert basis of a rational cone
• the lattice points of a lattice polytope
• the normalization of an affine monoid
• the h-vector and the Hilbert (Ehrhart) polynomial in the homogeneous case (for example, lattice polytopes)
• a description of the cone and lattice under consideration by a system of inequalities, equations and congruences

Normaliz can be started from the command line or from the GUI interface jNormaliz (written by Vinicius Almendra and Bogdan Ichim). jNormaliz is included in the distribution. See the Normaliz documentation for details.

The user indicates the type of input data in the input file and controls the computation and the output via the GUI interface or command line options.

Normaliz is provided for 2 degrees of integer precision: 64 bits or infinite. For infinite precision it uses the GMP (Linux, Mac) and MPIR (Windows) libraries. The user can require arithmetic checks at critical steps of the algorithms.

Normaliz comes with interfaces for Macaulay2 and Singular. The Macaulay2 interface (written by Gesa Kämpf) needs Macaulay2 1.1.99 or later. The Singular interface needs Singular 3-0-0 or later. Normaliz is accessible from polymake (thanks to an interface written by Andreas Paffenholz).

Some interesting and challenging examples document the power of Normaliz. Please send examples that you would like to add to the collection to one of the authors!

References to articles about Normaliz are included in the documentation and their pdf files can be found in the distribution.

Normaliz is distributed under GPL.

Current version: 2.5 (uploaded July 28, 2010) Previous version: 2.2

Installation:

1. Download the basic package below.

2. Download one of the executable packages below. (Windows users may have to install the Microsoft Visual C++ 2008 SP1 Redistributable Package; see below.)

3. Unzip both of them in a directory of your choice.

In this directory you will then find a directory Normaliz2.5 that contains the complete Normaliz system.

4. In the directory Normaliz 2.5 you will find jNormaliz.jar. You can run jNormaliz by clicking it in the appropriate way on your desktop.

5. If you want to run Normaliz from the command line, it may be necessary to copy norm64 and normbig to a directory in your search path.

Download:

Basic package Normaliz 2.5 source, documentation, examples, jNormaliz

This package contains documentation, examples, source code, and interfaces to Singular and Macaulay2. Please unzip the downloaded package in a directory of your choice. Because of their size, some example output files have not been included in the package. You can download ExtraOutput.zip separately.

Executables (to be unzipped in the same directory):

Apple (Intel)

Linux 32

Linux 64

The following Windows executables use OpenMP and therefore require the installation of the Microsoft Visual C++ 2008 SP1 Redistributable Package x86 (Windows 32) or the corresponding package x64 (Windows 64). (There is a good chance that the package exists on your machine. Just try to run norm64. If it starts, the package has already been installed.)

Windows 32

Windows 64

The following two executables should run without any preparations:

Windows 32 Serial

Windows 64 Serial

Note that it may be necessary to re-compile Normaliz on your system. Normaliz is compatible with the GNU C++ compiler (source included in the main package) and Microsoft Visual Studio. See the documentation for compilation details.

Visual C++ project

History: The first version of Normaliz was a C program developed by Winfried Bruns and Robert Koch in 1997/98. In 2002 Bruns added a Singular library. Witold Jarnicki improved the h-vector computation in 2003 and augmented the output options.

Version 2.0 (of 2008) was completely rewritten in C++ by Bogdan Ichim. The algorithms and their implementations were significantly improved by the authors. Moreover, additional input modes and user-controlled arithmetic checks were added.

Version 2.1 was augmented in regard to the user interface. Moreover, Pottier’s algorithm was added for solving systems of inequalities and equations.

Another addition was the Macaulay2 interface, implemented by Gesa Kämpf.

In version 2.2 the user interface was improved by Christof Söger.

Version 2.5 adds congruences, allowing the combination with equations and inequalities, and binomial ideals as further input types. Algorithms for large examples give access to hitherto unreachable cones. Various aspects, in particular the computation of shellings, have been improved considerably, and Normaliz can now exploits parallelization on shared memory systems via OpenMP. The main output file is re-organized.