Normaliz is an open source tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones.
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Normaliz - a tool for discrete convex geometry

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

Computation goals

  • 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 basis of rational, not necessarily pointed cones
  • normalization of affine monoids
  • lattice points of rational polytopes and (unbounded) polyhedra
  • 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

Normaliz offers the API libnormaliz that allows the user to access tNormaliz 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:

Sample input and output

The file 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. 2-dimensional cone

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:

The Python packages PyNormaliz and PyQNormaliz by Sebastian Gutsche provide an envirinment for interactive access. They are contained in the source package of Normaliz.

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


Optional packages

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

QNormaliz needs Flint, antic, arb and e-antic.


All files can be found at

Download and extract

  • the source basic package (or tar.gz) or the extended source package (x.y.z denotes the version) from the release page of this repository. (The full package contains jNormaliz and the Singular and Macaulay2 interfaces.)

Download and extract

  • the executable for your system (, or

Or compile Normaliz yourself on Linux or MacOS by one of the installation scripts

  • (only Normaliz)
  • (Normaliz and QNormaliz)

Docker image

available from