Normaliz is a open source tool for computations in affine monoids, vector configurations, rational polyhedra and rational cones. Normaliz now computes rational and algebraic polyhedra, i.e., polyhedra defined over real algebraic extensions of QQ.

• convex hulls and dual cones
• conversion from generators to constraints and vice versa
• projections of cones and polyhedra
• triangulations, disjoint decompositions and Stanley decompositions
• Hilbert bases of rational, not necessarily pointed cones
• normalizations of affine monoids
• lattice points of polytopes and (unbounded) polyhedra
• lattice points of polytopes satisfying polynomial constraints
• automorphism groups (euclidean, integral, rational/algebraic, combinatorial)
• face lattices and f-vectors
• euclidean and lattice normalized volumes of polytopes
• Hilbert (or Ehrhart) series and (quasi) polynomials under Z-gradings (for example, for rational polytopes)
• generalized (or weighted) Ehrhart series and Lebesgue integrals of - polynomials over rational polytopes
• minimal systems of generators and Hilbert series of general affine monoids
• Markov and Gröbner bases of lattice and toric ideals
• Special interface and functions for fusion rings (NEW)

Normaliz offers the API libnormaliz that allows the user to access Normaliz computations from any C++ program.

The frontend Normaliz reads input files and writes output files. There is a wide variety of input types to specify polyhedra and lattices by generators (vertices, extreme rays, bases) or by constraints (inequalities, equations and congruences). The user sets computation goals and chooses algorithmic variants through command line options or the input file.

Online exploration of Normaliz: https://mybinder.org/v2/gh/Normaliz/NormalizJupyter/master

The file 2cone.in from the directory example contains

amb_space 2
cone 2
1 3
2 1

It defines a cone in two-dimensional real space by its extreme rays.

The command

normaliz example/2cone

runs Normaliz with its default computation goals. It produces the output file 2cone.out (here typeset in two columns):

4 Hilbert basis elements          embedding dimension = 2
2 extreme rays                    rank = 2 (maximal)
2 support hyperplanes             external index = 1
                                  internal index = 5
                                  original monoid is not integrally closed

size of triangulation   = 1       rank of class group = 0
resulting sum of |det|s = 5       finite cyclic summands:
                                  5: 1
No implicit grading found

***********************************************************************

4 Hilbert basis elements:         2 extreme rays:
 1 1                               1 3
 1 2                               2 1
 1 3
 2 1                              2 support hyperplanes:
                                   -1  2
                                    3 -1

The main point was the computation of the Hilbert basis (encircled in red in the figure).

Each release contains executables for Linux 64, MacOS X and MS Windows 64.

Normaliz can be called from several other systems:

• CoCoA
• GAP
• Macaulay2
• polymake
• Singular
• SageMath

The Python package PyNormaliz by Sebastian Gutsche provides an environment for interactive access. It is contained in the source package of Normaliz.

jNormaliz by Vicinius Almendra and Bogdan Ichim provides a GUI to Normaliz

For its basic functionality Normaliz needs only GMP. Parallelization is based on OpenMP. For the computation of integrals CoCoALib is used.

For algebraic polyhedra Normaliz needs Flint and e-antic

The computation of automorphism groups uses nauty.

All files can be found at https://github.com/Normaliz/Normaliz/releases.

For the binary package (ready made executable program) download and extract

• the executable for your system (normaliz-x.y.zLinux64.zip, normaliz-x.y.zMacOS.zip or normaliz-x.y.zLinux64.zip).

For the source package download and extract

• normaliz-x.y.z.zip (or tar.gz)

From the source one can compile Normaliz oneself on Linux or MacOS by one of the installation scripts

• install_normaliz_with_opt.sh (only rational polyhedra)
• install_normaliz_with_eantic.sh (with algebraic polyhedra)

available from https://hub.docker.com/r/normaliz/normaliz/

Normaliz is available as a Debian, Gentoo and Ubuntu package, as well as from Conda (Linux, MacOS, MS Windows).