cone_property_wrappers.gd

#! @Chapter Functions
#! @Section Cone properties

#! @Arguments cone
#! @Returns the affine dimension
#! @Description
#! The affine dimension of the polyhedron in inhomogeneous computations. Its computation is triggered if necessary.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "AffineDim" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzAffineDim" );

#! @Arguments cone
#! @Returns the class group in a special format
#! @Description
#! 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 <M>n,m</M>. Such two entries
#! represent a free cyclic summand <M>(\mathbb{Z}/n\mathbb{Z})^m</M>.
#! Not allowed in inhomogeneous computations.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "ClassGroup" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzClassGroup" );

#! @Arguments cone
#! @Returns a matrix whose rows represent the congruences
#! @Description
#! The equations, congruences and support hyperplanes together
#! describe the lattice points of the cone.
#! <P/>
#! This is part of the cone property <Q>Sublattice</Q>.
DeclareGlobalFunction( "NmzCongruences" );

#! @Arguments cone
#! @Returns a matrix whose rows are the degree 1 elements
#! @Description
#! Requires the presence of a grading. Not allowed in inhomogeneous computations.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "Deg1Elements" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzDeg1Elements" );

#! @Arguments cone
#! @Returns the dehomgenization vector
#! @Description
#! Only for inhomogeneous computations.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "Dehomogenization" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzDehomogenization" );

#! @Arguments cone
#! @Returns the embedding dimension of the cone
#! @Description
#! 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.
DeclareGlobalFunction( "NmzEmbeddingDimension" );

#! @Arguments cone
#! @Returns a matrix whose rows represent the equations
#! @Description
#! 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.
DeclareGlobalFunction( "NmzEquations" );

#! @Arguments cone
#! @Returns a matrix whose rows represent the excluded faces
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "ExcludedFaces" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzExcludedFaces" );

#! @Arguments cone
#! @Returns a matrix whose rows are the extreme rays
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "ExtremeRays" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzExtremeRays" );

#! @Arguments cone
#! @Returns a matrix whose rows are the generators of <A>cone</A>
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "Generators" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzGenerators" );

#! @Arguments cone
#! @Returns a vector representing the generator of the interior of <A>cone</A>
#! @Description
#! If <A>cone</A> is Gorenstein, this function returns the generator of the interior of <A>cone</A>.
#! If <A>cone</A> is not Gorenstein, an error is raised.
DeclareGlobalFunction( "NmzGeneratorOfInterior" );

#! @Arguments cone
#! @Returns the grading vector
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "Grading" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzGrading" );

#! @Arguments cone
#! @Returns a matrix whose rows are the Hilbert basis elements
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "HilbertBasis" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzHilbertBasis" );

#! @Arguments cone
#! @Returns the Hilbert function as a quasipolynomial
#! @Description
#! The Hilbert function counts the lattice points degreewise. The result is a
#! quasipolynomial <M>Q</M>, that is, a polynomial with periodic coefficients. It is
#! given as list of polynomials <M>P_0, \ldots, P_{(p-1)}</M> such that <M>Q(i) = P_{(i \bmod p)} (i)</M>.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "HilbertQuasiPolynomial" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzHilbertQuasiPolynomial" );

#! @Arguments cone
#! @Returns the Hilbert series as rational function
#! @Description
#! 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 <M>[k_i, l_i]</M>. Such a pair represents the factor <M>(1-t^{k_i})^{l_i}</M>.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "HilbertSeries" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzHilbertSeries" );

#! @Arguments cone
#! @Returns inclusion-exclusion data
#! @Description
#! 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 <Ref Func="NmzTriangulation"/>) and the
#! multiplicity with which it was considered.
#! Only available with excluded faces or strict constraints as input.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "InclusionExclusionData" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzInclusionExclusionData" );

#! @Arguments cone
#! @Returns <K>true</K> if all extreme rays have degree 1; <K>false</K> otherwise
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "IsDeg1ExtremeRays" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzIsDeg1ExtremeRays" );

#! @Arguments cone
#! @Returns <K>true</K> if all Hilbert basis elements have degree 1; <K>false</K> otherwise
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "IsDeg1HilbertBasis" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzIsDeg1HilbertBasis" );

#! @Arguments cone
#! @Returns whether the cone is Gorenstein
#! @Description
#! Returns true if <A>cone</A> is Gorenstein, false otherwise.
DeclareGlobalFunction( "NmzIsGorenstein" );

#! @Arguments cone
#! @Returns whether the cone is inhomogeneous
#! @Description
#! This value is always known directly after the creation of the cone.
DeclareGlobalFunction( "NmzIsInhomogeneous" );

#! @Arguments cone
#! @Returns <K>true</K> if the cone is integrally closed; <K>false</K> otherwise
#! @Description
#! 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.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "IsIntegrallyClosed" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzIsIntegrallyClosed" );

#! @Arguments cone
#! @Returns <K>true</K> if the cone is pointed; <K>false</K> otherwise
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "IsPointed" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzIsPointed" );

#! @Arguments cone
#! @Returns <K>true</K> if is the monomial ideal is primary to the irrelevant maximal ideal, <K>false</K> otherwise
#! @Description
#! Only used with the input type <C>rees_algebra</C>.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "IsReesPrimary" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzIsReesPrimary" );

#! @Arguments cone
#! @Returns a matrix whose rows generate the maximale linear subspace
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "MaximalSubspace" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzMaximalSubspace" );

#! @Arguments cone
#! @Returns a matrix whose rows are the module generators
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "ModuleGenerators" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzModuleGenerators" );

#! @Arguments cone
#! @Returns a matrix whose rows are the module generators over the original monoid
#! @Description
#! 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.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "ModuleGeneratorsOverOriginalMonoid" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzModuleGeneratorsOverOriginalMonoid" );

#! @Arguments cone
#! @Returns the rank of the module of lattice points in the polyhedron as a module over the recession monoid
#! @Description
#! Only for inhomogeneous computations.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "ModuleRank" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzModuleRank" );

#! @Arguments cone
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "Multiplicity" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzMultiplicity" );

#! @Arguments cone
#! @Returns a matrix whose rows are the original monoid generators
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "OriginalMonoidGenerators" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzOriginalMonoidGenerators" );

#! @Arguments cone
#! @Returns the rank of the cone
#! @Description
#! This value is the rank of the lattice generated by the lattice points of the cone.
#! <P/>
#! This is part of the cone property <Q>Sublattice</Q>.
DeclareGlobalFunction( "NmzRank" );

#! @Arguments cone
#! @Returns the rank of the recession cone
#! @Description
#! Only for inhomogeneous computations.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "RecessionRank" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzRecessionRank" );

#! @Arguments cone
#! @Description
#! the multiplicity of a monomial ideal, provided it is primary to the maximal
#! ideal generated by the indeterminates. Used only with the input type
#! <C>rees_algebra</C>.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "ReesPrimaryMultiplicity" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzReesPrimaryMultiplicity" );

#! @Arguments cone
#! @Returns a matrix whose rows represent the support hyperplanes
#! @Description
#! 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.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "SupportHyperplanes" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzSupportHyperplanes" );

#! @Arguments cone
#! @Returns the triangulation
#! @Description
#! 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 <Ref Func="NmzGenerators"/> (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.
#! <P/>
#! This is an alias for <C>NmzConeProperty( cone, "Triangulation" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzTriangulation" );

#! @Arguments cone
#! @Returns sum of the absolute values of the determinants of the simplicial cones in the used triangulation
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "TriangulationDetSum" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzTriangulationDetSum" );

#! @Arguments cone
#! @Returns the number of simplicial cones in the used triangulation
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "TriangulationSize" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzTriangulationSize" );

#! @Arguments cone
#! @Returns a matrix whose rows are the vertices of the polyhedron <A>cone</A> with float coordinates
#! @Description
#! The rows of this matrix represent the vertices of <A>cone</A>, printed as floats for better readability.
#! The result might be inexact, and should therefore not be used for computations.
DeclareGlobalFunction( "NmzVerticesFloat" );

#! @Arguments cone
#! @Returns a matrix whose rows are the vertices of the polyhedron
#! @Description
#! This is an alias for <C>NmzConeProperty( cone, "VerticesOfPolyhedron" );</C> see <Ref Func="NmzConeProperty"/>.
DeclareGlobalFunction( "NmzVerticesOfPolyhedron" );


#! @Description
#!  This is an alias for NmzConeProperty( cone, "ConeDecomposition" );
#! @Arguments cone
DeclareGlobalFunction( "NmzConeDecomposition" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "EmbeddingDim" );
#! @Arguments cone
DeclareGlobalFunction( "NmzEmbeddingDim" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "ExternalIndex" );
#! @Arguments cone
DeclareGlobalFunction( "NmzExternalIndex" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "GradingDenom" );
#! @Arguments cone
DeclareGlobalFunction( "NmzGradingDenom" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "IntegerHull" );
#! @Arguments cone
DeclareGlobalFunction( "NmzIntegerHull" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "InternalIndex" );
#! @Arguments cone
DeclareGlobalFunction( "NmzInternalIndex" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "StanleyDec" );
#! @Arguments cone
DeclareGlobalFunction( "NmzStanleyDec" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "Sublattice" );
#! @Arguments cone
DeclareGlobalFunction( "NmzSublattice" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "UnitGroupIndex" );
#! @Arguments cone
DeclareGlobalFunction( "NmzUnitGroupIndex" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartQuasiPolynomial" );
#! @Arguments cone
DeclareGlobalFunction( "NmzWeightedEhrhartQuasiPolynomial" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartSeries" );
#! @Arguments cone
DeclareGlobalFunction( "NmzWeightedEhrhartSeries" );

#! @Description
#!  This is an alias for NmzConeProperty( cone, "WitnessNotIntegrallyClosed" );
#! @Arguments cone
DeclareGlobalFunction( "NmzWitnessNotIntegrallyClosed" );