**Normaliz**

Winfried Bruns, Bogdan Ichim, Tim Römer and Christof Söger

The Normaliz project is supported by the DFG
SPP 1489

"Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie"

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 (or lexicographic)
triangulation of a vector configuration (resulting in a triangulation of
the cone generated by it)
- the Hilbert basis of a rational
cone
- the lattice points of a
rational polytope
- NEW in 2.11: the lattice points of a
rational (unbounded) polyhedron
- the normalization of an affine
monoid
- the Hilbert (or Ehrhart) series and the Hilbert (or
Ehrhart) (quasi) polynomial under a Z-grading (for example, for rational
polytopes)
- NEW in 2.11: Ehrhart series for semiopen
cones
- generalized (or weighted)
Ehrhart series and Lebesgue integrals of polynomials over rational
polytopes via NmzIntegrate
(also in the semiopen case)
- 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.

NmzIntegrate
is based on CoCoALib.

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 the polymake team) and is used by B. Burton's

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:** 2.11.2 (including
NmzIntegrate 1.2) (uploaded August 1, 2014) Previous versions: 2.10.1 2.8 2.7 2.5

Version
2.11.2 takes care of several shortcomings of 2.11.1 in computations with
inhomogeneous input.