# Normaliz

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Computer algebra system

Normaliz Original authors Winfried Bruns, Robert Koch, Bogdan Ichim, Christof Soeger Stable release 3.11.1[1] / 2 February 2026; 4 months ago (2 February 2026) Written in C++, C and Python Type Computer algebra system License GNU General Public License, version 3 Website normaliz.uni-osnabrueck.de Repository github.com/normaliz/Normaliz

**Normaliz** is a [free](/source/Free_software) [computer algebra system](/source/Computer_algebra_system) developed by Winfried Bruns, Robert Koch (1998–2002), Bogdam Ichim (2007/08) and Christof Soeger (2009–2016). It is published under the [GNU General Public License](/source/GNU_General_Public_License) version 2.

Normaliz computes lattice points in rational [polyhedra](/source/Polyhedra), or, in other terms, solves [linear diophantine systems of equations](/source/Diophantine_equation#System_of_linear_Diophantine_equations), inequalities, and congruences. Special tasks are the computation of lattice points in bounded rational polytopes and [Hilbert bases](/source/Hilbert_basis_(linear_programming)) of rational cones. Normaliz also computes enumerative data, such as multiplicities (volumes) and [Hilbert series](/source/Hilbert_series). The kernel of Normaliz is a templated C++ class library. For multivariate polynomial arithmetic it uses [CoCoALib](/source/CoCoA).

Normaliz has interfaces to several general computer algebra systems: [CoCoA](/source/CoCoA), [GAP](/source/GAP_(computer_algebra_system)), [Macaulay2](/source/Macaulay2) and [Singular](/source/Singular_(software)). It can be used interactively via its Python interface PyNormaliz. Its use in SageMath is in preparation.

[Jesús A. De_Loera](/source/Jes%C3%BAs_A._De_Loera) cites Normaliz among his favorite programs for computing [Hilbert basis](/source/Hilbert_basis_(linear_programming)).[2]

## See also

- [Free and open-source software portal](https://en.wikipedia.org/wiki/Portal:Free_and_open-source_software)

- [Comparison of computer algebra systems](/source/Comparison_of_computer_algebra_systems)

## References

1. **[^](#cite_ref-wikidata-e729f3a8643835cf7407b1150ed3b8d643bd6821-v20_1-0)** ["Release 3.11.1"](https://github.com/Normaliz/Normaliz/releases/tag/v3.11.1). 2 February 2026. Retrieved 3 April 2026.

1. **[^](#cite_ref-2)** De Loera, Jesús. "Combinatorial Problems with Geometric Solutions". [*Course Notes: Algebraic and Geometric Combinatorics*](https://www.math.ucdavis.edu/~deloera/TEACHING/MATH246/NOTES/chapter2.tex). UC Davis.

## External links

- [Official website](https://www.normaliz.uni-osnabrueck.de/)

- [Publications and examples of Normaliz applications](https://www.normaliz.uni-osnabrueck.de/documentation/)

- [http://github.com/normaliz/Normaliz](https://github.com/normaliz/Normaliz)

v t e Computer algebra systems Open-source Axiom Cadabra CoCoA Fermat FriCAS FORM GAP GiNaC Macaulay2 Maxima Normaliz PARI/GP Reduce SageMath Singular SymPy Xcas/Giac Yacas Proprietary ClassPad Manager Engineering Equation Solver KANT Magma Maple Mathcad muPAD (MATLAB symbolic math toolbox) SMath Studio TI InterActive! Wolfram Mathematica Discontinued CAMAL Derive Erable LiveMath Macsyma Mathomatic muMATH ALTRAN Category List

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Adapted from the Wikipedia article [Normaliz](https://en.wikipedia.org/wiki/Normaliz) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normaliz?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.