is an open source tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones.

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 the Normaliz computations from any C++ program.

Online exploration of Normaliz:


The Normaliz project was partially supported by the DFG SPP 1489 “Algorithmische und experimentelle Methoden in Algebra, Geometrie und Zahlentheorie”.