Functions

In this chapter we describe the functions offered by NormalizInterface. All functions supplied by this package start with Nmz. For examples see the chapter .

Create a NmzCone

To create a cone object use .

NmzCone Creates a NmzCone. The list argument should contain an even number of elements, alternating between a string and a integer matrix. The string has to correspond to a Normaliz input type string and the following matrix will be interpreted as input of that type.

Currently the following strings are recognized: integral_closure, polyhedron, normalization, polytope, rees_algebra, inequalities, strict_inequalities, signs, strict_signs, equations, congruences, inhom_inequalities, inhom_equations, inhom_congruences, dehomogenization, lattice_ideal, grading, excluded_faces, lattice, saturation, cone, offset, vertices, support_hyperplanes, cone_and_lattice, subspace.

See the Normaliz manual for a detailed description.

<#Include Label="NmzCone_example">
Use a NmzCone

whether the cone has already computed the given property See for a list of recognized properties.

<#Include Label="NmzHasConeProperty_example"> a list of strings representing the known (computed) cone properties Given a Normaliz cone object, return a list of all properties already computed for the cone.

<#Include Label="NmzKnownConeProperties_example"> the previous verbosity Set the global default verbosity state in libnormaliz. This will influence all NmzCone created afterwards, but not any existing ones.

See also the previous verbosity Set the verbosity state for a cone.

See also a boolean indicating success Start computing properties of the given cone. The first parameter indicates a cone object, the second parameter is either a single string, or a list of strings, which indicate what should be computed.

The single parameter version is equivalent to NmzCone(cone, ["DefaultMode"]). See for a list of recognized properties. <#Include Label="NmzCompute_example"> the result of the computation, type depends on the property Triggers the computation of the property of the cone and returns the result. If the property was already known, it is not recomputed. Currently the following strings are recognized as properties: Generators see , ExtremeRays see , VerticesOfPolyhedron see , SupportHyperplanes see , TriangulationSize see , TriangulationDetSum see , Triangulation see , Multiplicity see , RecessionRank see , AffineDim see , ModuleRank see , HilbertBasis see , ModuleGenerators see , Deg1Elements see , HilbertSeries see , HilbertQuasiPolynomial see , Grading see , IsPointed see , IsDeg1ExtremeRays see , IsDeg1HilbertBasis see , IsIntegrallyClosed see , OriginalMonoidGenerators see , IsReesPrimary see , ReesPrimaryMultiplicity see , ExcludedFaces see , Dehomogenization see , InclusionExclusionData see , ClassGroup see , ModuleGeneratorsOverOriginalMonoid see , Sublattice computes the efficient sublattice and returns a bool signaling whether the computation was successful. Actual data connected to it can be accessed by , , , and .

Additionally also the following compute options are accepted as property. They modify what and how should be computed, and return True after a successful computation. Approximate approximate the rational polytope by an integral polytope, currently only useful in combination with Deg1Elements. BottomDecomposition use the best possible triangulation (with respect to the sum of determinants) using the given generators. DefaultMode try to compute what is possible and do not throw an exception when something cannot be computed. DualModeactivates the dual algorithm for the computation of the Hilbert basis and degree 1 elements. Includes HilbertBasis, unless Deg1Elements is set. Often a good choice if you start from constraints. KeepOrder forbids to reorder the generators. Blocks BottomDecomposition.

All the properties above can be given to . There you can combine different properties, e.g. give some properties that you would like to know and add some compute options.

See the Normaliz manual for a detailed description.

Print an overview of all known properties of the given cone, as well as their values.
Cone properties

the affine dimension The affine dimension of the polyhedron in inhomogeneous computations. Its computation is triggered if necessary.

This is an alias for NmzConeProperty( cone, "AffineDim" ); see . the class group in a special format A normal affine monoid M has a well-defined divisor class group. It is naturally isomorphic to the divisor class group of K[M] where K is a field (or any unique factorization domain). We represent it as a vector where the first entry is the rank. It is followed by sequence of pairs of entries n,m. Such two entries represent a free cyclic summand (\mathbb{Z}/n\mathbb{Z})^m. Not allowed in inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "ClassGroup" ); see . a matrix whose rows represent the congruences The equations, congruences and support hyperplanes together describe the lattice points of the cone.

This is part of the cone property Sublattice. a matrix whose rows are the degree 1 elements Requires the presence of a grading. Not allowed in inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "Deg1Elements" ); see . the dehomgenization vector Only for inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "Dehomogenization" ); see . the embedding dimension of the cone The embedding dimension is the dimension of the space in which the computation is done. It is the number of components of the output vectors. This value is always known directly after the creation of the cone. a matrix whose rows represent the equations The equations cut out the linear space generated by the cone. The equations, congruences and support hyperplanes together describe the lattice points of the cone. a matrix whose rows represent the excluded faces This is an alias for NmzConeProperty( cone, "ExcludedFaces" ); see . a matrix whose rows are the extreme rays This is an alias for NmzConeProperty( cone, "ExtremeRays" ); see . a matrix whose rows are the generators of cone This is an alias for NmzConeProperty( cone, "Generators" ); see . a vector representing the generator of the interior of cone If cone is Gorenstein, this function returns the generator of the interior of cone. If cone is not Gorenstein, an error is raised. the grading vector This is an alias for NmzConeProperty( cone, "Grading" ); see . a matrix whose rows are the Hilbert basis elements This is an alias for NmzConeProperty( cone, "HilbertBasis" ); see . the Hilbert function as a quasipolynomial The Hilbert function counts the lattice points degreewise. The result is a quasipolynomial Q, that is, a polynomial with periodic coefficients. It is given as list of polynomials P_0, \ldots, P_{(p-1)} such that Q(i) = P_{(i \bmod p)} (i).

This is an alias for NmzConeProperty( cone, "HilbertQuasiPolynomial" ); see . the Hilbert series as rational function The result consists of a list with two entries. The first is the numerator polynomial. In inhomogeneous computations this can also be a Laurent polynomial. The second list entry represents the denominator. It is a list of pairs [k_i, l_i]. Such a pair represents the factor (1-t^{k_i})^{l_i}.

This is an alias for NmzConeProperty( cone, "HilbertSeries" ); see . inclusion-exclusion data List of faces which are internally have been used in the inclusion-exclusion scheme. Given as a list pairs. The first pair entry is a key of generators contained in the face (compare also ) and the multiplicity with which it was considered. Only available with excluded faces or strict constraints as input.

This is an alias for NmzConeProperty( cone, "InclusionExclusionData" ); see . true if all extreme rays have degree 1; false otherwise This is an alias for NmzConeProperty( cone, "IsDeg1ExtremeRays" ); see . true if all Hilbert basis elements have degree 1; false otherwise This is an alias for NmzConeProperty( cone, "IsDeg1HilbertBasis" ); see . whether the cone is Gorenstein Returns true if cone is Gorenstein, false otherwise. whether the cone is inhomogeneous This value is always known directly after the creation of the cone. true if the cone is integrally closed; false otherwise It is integrally closed when the Hilbert basis is a subset of the original monoid generators. So it is only computable if we have original monoid generators.

This is an alias for NmzConeProperty( cone, "IsIntegrallyClosed" ); see . true if the cone is pointed; false otherwise This is an alias for NmzConeProperty( cone, "IsPointed" ); see . true if is the monomial ideal is primary to the irrelevant maximal ideal, false otherwise Only used with the input type rees_algebra.

This is an alias for NmzConeProperty( cone, "IsReesPrimary" ); see . a matrix whose rows generate the maximale linear subspace This is an alias for NmzConeProperty( cone, "MaximalSubspace" ); see . a matrix whose rows are the module generators This is an alias for NmzConeProperty( cone, "ModuleGenerators" ); see . a matrix whose rows are the module generators over the original monoid A minimal system of generators of the integral closure over the original monoid. Requires the existence of original monoid generators. Not allowed in inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "ModuleGeneratorsOverOriginalMonoid" ); see . the rank of the module of lattice points in the polyhedron as a module over the recession monoid Only for inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "ModuleRank" ); see . This is an alias for NmzConeProperty( cone, "Multiplicity" ); see . a matrix whose rows are the original monoid generators This is an alias for NmzConeProperty( cone, "OriginalMonoidGenerators" ); see . the rank of the cone This value is the rank of the lattice generated by the lattice points of the cone.

This is part of the cone property Sublattice. the rank of the recession cone Only for inhomogeneous computations.

This is an alias for NmzConeProperty( cone, "RecessionRank" ); see . the multiplicity of a monomial ideal, provided it is primary to the maximal ideal generated by the indeterminates. Used only with the input type rees_algebra.

This is an alias for NmzConeProperty( cone, "ReesPrimaryMultiplicity" ); see . a matrix whose rows represent the support hyperplanes The equations cut out the linear space generated by the cone. The equations, congruences and support hyperplanes together describe the lattice points of the cone.

This is an alias for NmzConeProperty( cone, "SupportHyperplanes" ); see . the triangulation This returns a list of the maximal simplicial cones in a triangulation, i.e., a list of cones dividing the cone into simplicial cones. Each cone in the list is represented by a pair. The first entry of such a pair is the key of the simplex, i.e., a list of integers a_1,\dots,a_n referring to the (counting from 0) which are used in this simplicial cone. The second entry of each pair in the list is the absolute value of the determinant of the generator matrix of the simplicial cone.

This is an alias for NmzConeProperty( cone, "Triangulation" ); see . sum of the absolute values of the determinants of the simplicial cones in the used triangulation This is an alias for NmzConeProperty( cone, "TriangulationDetSum" ); see . the number of simplicial cones in the used triangulation This is an alias for NmzConeProperty( cone, "TriangulationSize" ); see . a matrix whose rows are the vertices of the polyhedron cone with float coordinates The rows of this matrix represent the vertices of cone, printed as floats for better readability. The result might be inexact, and should therefore not be used for computations. a matrix whose rows are the vertices of the polyhedron This is an alias for NmzConeProperty( cone, "VerticesOfPolyhedron" ); see . This is an alias for NmzConeProperty( cone, "ConeDecomposition" ); see . This is an alias for NmzConeProperty( cone, "EmbeddingDim" ); see . This is an alias for NmzConeProperty( cone, "ExternalIndex" ); see . This is an alias for NmzConeProperty( cone, "GradingDenom" ); see . This is an alias for NmzConeProperty( cone, "IntegerHull" ); see . This is an alias for NmzConeProperty( cone, "InternalIndex" ); see . This is an alias for NmzConeProperty( cone, "StanleyDec" ); see . This is an alias for NmzConeProperty( cone, "Sublattice" ); see . This is an alias for NmzConeProperty( cone, "UnitGroupIndex" ); see . This is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartQuasiPolynomial" ); see . This is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartSeries" ); see . This is an alias for NmzConeProperty( cone, "WitnessNotIntegrallyClosed" ); see . This is an alias for NmzConeProperty( cone, "EhrhartSeries" ); see .

Supported in Normaliz >= 3.5.0. This is an alias for NmzConeProperty( cone, "EuclideanVolume" ); see .

Supported in Normaliz >= 3.5.0. This is an alias for NmzConeProperty( cone, "ProjectCone" ); see .

Supported in Normaliz >= 3.5.0. This is an alias for NmzConeProperty( cone, "SuppHypsFloat" ); see .

Supported in Normaliz >= 3.5.2. This is an alias for NmzConeProperty( cone, "Volume" ); see .

Supported in Normaliz >= 3.5.0. This is an alias for NmzConeProperty( cone, "FVector" ); see .

Supported in Normaliz >= 3.7.0. This is an alias for NmzConeProperty( cone, "FaceLattice" ); see .

Supported in Normaliz >= 3.7.0. a record describing the basis change The result record r has three components: r.Embedding, r.Projection, and r.Annihilator, where the embedding A and the projection B are matrices, and the annihilator c is an integer. They represent the mapping into the effective lattice \mathbb{Z}^n \to \mathbb{Z}^r, u \mapsto (uB)/c and the inverse operation \mathbb{Z}^r \to \mathbb{Z}^n, v \mapsto vA.

This is part of the cone property Sublattice.