  
  [1X2 [33X[0;0YFunctions[133X[101X
  
  [33X[0;0YIn  this chapter we describe the functions offered by [3XNormalizInterface[103X. All
  functions  supplied  by  this  package  start with [21XNmz[121X. For examples see the
  chapter [14X'[33X[0;0YExamples[133X'[114X.[133X
  
  
  [1X2.1 [33X[0;0YCreate a NmzCone[133X[101X
  
  [33X[0;0YTo create a cone object use [3XNmzCone[103X.[133X
  
  [1X2.1-1 NmzCone[101X
  
  [29X[2XNmzCone[102X( [3Xlist[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YNmzCone[133X
  
  [33X[0;0YCreates  a  NmzCone.  The  [3Xlist[103X  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.[133X
  
  [33X[0;0YCurrently the following strings are recognized:[133X
  
  [30X    [33X[0;6Y[10Xintegral_closure[110X,[133X
  
  [30X    [33X[0;6Y[10Xpolyhedron[110X,[133X
  
  [30X    [33X[0;6Y[10Xnormalization[110X,[133X
  
  [30X    [33X[0;6Y[10Xpolytope[110X,[133X
  
  [30X    [33X[0;6Y[10Xrees_algebra[110X,[133X
  
  [30X    [33X[0;6Y[10Xinequalities[110X,[133X
  
  [30X    [33X[0;6Y[10Xstrict_inequalities[110X,[133X
  
  [30X    [33X[0;6Y[10Xsigns[110X,[133X
  
  [30X    [33X[0;6Y[10Xstrict_signs[110X,[133X
  
  [30X    [33X[0;6Y[10Xequations[110X,[133X
  
  [30X    [33X[0;6Y[10Xcongruences[110X,[133X
  
  [30X    [33X[0;6Y[10Xinhom_inequalities[110X,[133X
  
  [30X    [33X[0;6Y[10Xinhom_equations[110X,[133X
  
  [30X    [33X[0;6Y[10Xinhom_congruences[110X,[133X
  
  [30X    [33X[0;6Y[10Xdehomogenization[110X,[133X
  
  [30X    [33X[0;6Y[10Xlattice_ideal[110X,[133X
  
  [30X    [33X[0;6Y[10Xgrading[110X,[133X
  
  [30X    [33X[0;6Y[10Xexcluded_faces[110X,[133X
  
  [30X    [33X[0;6Y[10Xlattice[110X,[133X
  
  [30X    [33X[0;6Y[10Xsaturation[110X,[133X
  
  [30X    [33X[0;6Y[10Xcone[110X,[133X
  
  [30X    [33X[0;6Y[10Xoffset[110X,[133X
  
  [30X    [33X[0;6Y[10Xvertices[110X,[133X
  
  [30X    [33X[0;6Y[10Xsupport_hyperplanes[110X,[133X
  
  [30X    [33X[0;6Y[10Xcone_and_lattice[110X,[133X
  
  [30X    [33X[0;6Y[10Xsubspace[110X.[133X
  
  [33X[0;0YSee the Normaliz manual for a detailed description.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcone := NmzCone(["integral_closure",[[2,1],[1,3]]]);[127X[104X
    [4X[28X<a Normaliz cone>[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YUse a NmzCone[133X[101X
  
  [1X2.2-1 NmzHasConeProperty[101X
  
  [29X[2XNmzHasConeProperty[102X( [3Xcone[103X, [3Xproperty[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ywhether the cone has already computed the given property[133X
  
  [33X[0;0YSee [2XNmzConeProperty[102X ([14X2.2-6[114X) for a list of recognized properties.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNmzHasConeProperty(cone, "ExtremeRays");[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-2 NmzKnownConeProperties[101X
  
  [29X[2XNmzKnownConeProperties[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya   list   of  strings  representing  the  known  (computed)  cone
            properties[133X
  
  [33X[0;0YGiven  a  Normaliz  cone  object,  return  a  list of all properties already
  computed for the cone.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNmzKnownConeProperties(cone);[127X[104X
    [4X[28X[ "Generators", "OriginalMonoidGenerators", "Sublattice" ][128X[104X
  [4X[32X[104X
  
  [1X2.2-3 NmzSetVerboseDefault[101X
  
  [29X[2XNmzSetVerboseDefault[102X( [3XverboseFlag[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe previous verbosity[133X
  
  [33X[0;0YSet  the  global default verbosity state in libnormaliz. This will influence
  all NmzCone created afterwards, but not any existing ones.[133X
  
  [33X[0;0YSee also [2XNmzSetVerbose[102X ([14X2.2-4[114X)[133X
  
  [1X2.2-4 NmzSetVerbose[101X
  
  [29X[2XNmzSetVerbose[102X( [3Xcone[103X, [3XverboseFlag[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe previous verbosity[133X
  
  [33X[0;0YSet the verbosity state for a cone.[133X
  
  [33X[0;0YSee also [2XNmzSetVerboseDefault[102X ([14X2.2-3[114X)[133X
  
  [1X2.2-5 NmzCompute[101X
  
  [29X[2XNmzCompute[102X( [3Xcone[103X[, [3Xpropnames[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ya boolean indicating success[133X
  
  [33X[0;0YStart  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.[133X
  
  [33X[0;0YThe    single    parameter    version   is   equivalent   to   [10XNmzCone(cone,
  ["DefaultMode"])[110X.  See  [2XNmzConeProperty[102X  ([14X2.2-6[114X)  for  a  list of recognized
  properties.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNmzKnownConeProperties(cone);[127X[104X
    [4X[28X[ "Generators", "OriginalMonoidGenerators", "Sublattice" ][128X[104X
    [4X[25Xgap>[125X [27XNmzCompute(cone, ["SupportHyperplanes", "IsPointed"]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XNmzKnownConeProperties(cone);[127X[104X
    [4X[28X[ "Generators", "ExtremeRays", "SupportHyperplanes", "IsPointed",[128X[104X
    [4X[28X  "IsDeg1ExtremeRays", "OriginalMonoidGenerators", "Sublattice",[128X[104X
    [4X[28X  "MaximalSubspace" ][128X[104X
    [4X[25Xgap>[125X [27XNmzCompute(cone);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XNmzKnownConeProperties(cone);[127X[104X
    [4X[28X[ "Generators", "ExtremeRays", "SupportHyperplanes", "TriangulationSize",[128X[104X
    [4X[28X  "TriangulationDetSum", "HilbertBasis", "IsPointed", "IsDeg1ExtremeRays",[128X[104X
    [4X[28X  "IsIntegrallyClosed", "OriginalMonoidGenerators", "Sublattice",[128X[104X
    [4X[28X  "ClassGroup", "MaximalSubspace"][128X[104X
  [4X[32X[104X
  
  [1X2.2-6 NmzConeProperty[101X
  
  [29X[2XNmzConeProperty[102X( [3Xcone[103X, [3Xproperty[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe result of the computation, type depends on the property[133X
  
  [33X[0;0YTriggers 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:[133X
  
  [30X    [33X[0;6Y[10XGenerators[110X see [2XNmzGenerators[102X ([14X2.3-10[114X),[133X
  
  [30X    [33X[0;6Y[10XExtremeRays[110X see [2XNmzExtremeRays[102X ([14X2.3-9[114X),[133X
  
  [30X    [33X[0;6Y[10XVerticesOfPolyhedron[110X see [2XNmzVerticesOfPolyhedron[102X ([14X2.3-38[114X),[133X
  
  [30X    [33X[0;6Y[10XSupportHyperplanes[110X see [2XNmzSupportHyperplanes[102X ([14X2.3-33[114X),[133X
  
  [30X    [33X[0;6Y[10XTriangulationSize[110X see [2XNmzTriangulationSize[102X ([14X2.3-36[114X),[133X
  
  [30X    [33X[0;6Y[10XTriangulationDetSum[110X see [2XNmzTriangulationDetSum[102X ([14X2.3-35[114X),[133X
  
  [30X    [33X[0;6Y[10XTriangulation[110X see [2XNmzTriangulation[102X ([14X2.3-34[114X),[133X
  
  [30X    [33X[0;6Y[10XMultiplicity[110X see [2XNmzMultiplicity[102X ([14X2.3-28[114X),[133X
  
  [30X    [33X[0;6Y[10XRecessionRank[110X see [2XNmzRecessionRank[102X ([14X2.3-31[114X),[133X
  
  [30X    [33X[0;6Y[10XAffineDim[110X see [2XNmzAffineDim[102X ([14X2.3-1[114X),[133X
  
  [30X    [33X[0;6Y[10XModuleRank[110X see [2XNmzModuleRank[102X ([14X2.3-27[114X),[133X
  
  [30X    [33X[0;6Y[10XHilbertBasis[110X see [2XNmzHilbertBasis[102X ([14X2.3-13[114X),[133X
  
  [30X    [33X[0;6Y[10XModuleGenerators[110X see [2XNmzModuleGenerators[102X ([14X2.3-25[114X),[133X
  
  [30X    [33X[0;6Y[10XDeg1Elements[110X see [2XNmzDeg1Elements[102X ([14X2.3-4[114X),[133X
  
  [30X    [33X[0;6Y[10XHilbertSeries[110X see [2XNmzHilbertSeries[102X ([14X2.3-15[114X),[133X
  
  [30X    [33X[0;6Y[10XHilbertQuasiPolynomial[110X see [2XNmzHilbertQuasiPolynomial[102X ([14X2.3-14[114X),[133X
  
  [30X    [33X[0;6Y[10XGrading[110X see [2XNmzGrading[102X ([14X2.3-12[114X),[133X
  
  [30X    [33X[0;6Y[10XIsPointed[110X see [2XNmzIsPointed[102X ([14X2.3-22[114X),[133X
  
  [30X    [33X[0;6Y[10XIsDeg1ExtremeRays[110X see [2XNmzIsDeg1ExtremeRays[102X ([14X2.3-17[114X),[133X
  
  [30X    [33X[0;6Y[10XIsDeg1HilbertBasis[110X see [2XNmzIsDeg1HilbertBasis[102X ([14X2.3-18[114X),[133X
  
  [30X    [33X[0;6Y[10XIsIntegrallyClosed[110X see [2XNmzIsIntegrallyClosed[102X ([14X2.3-21[114X),[133X
  
  [30X    [33X[0;6Y[10XOriginalMonoidGenerators[110X see [2XNmzOriginalMonoidGenerators[102X ([14X2.3-29[114X),[133X
  
  [30X    [33X[0;6Y[10XIsReesPrimary[110X see [2XNmzIsReesPrimary[102X ([14X2.3-23[114X),[133X
  
  [30X    [33X[0;6Y[10XReesPrimaryMultiplicity[110X see [2XNmzReesPrimaryMultiplicity[102X ([14X2.3-32[114X),[133X
  
  [30X    [33X[0;6Y[10XExcludedFaces[110X see [2XNmzExcludedFaces[102X ([14X2.3-8[114X),[133X
  
  [30X    [33X[0;6Y[10XDehomogenization[110X see [2XNmzDehomogenization[102X ([14X2.3-5[114X),[133X
  
  [30X    [33X[0;6Y[10XInclusionExclusionData[110X see [2XNmzInclusionExclusionData[102X ([14X2.3-16[114X),[133X
  
  [30X    [33X[0;6Y[10XClassGroup[110X see [2XNmzClassGroup[102X ([14X2.3-2[114X),[133X
  
  [30X    [33X[0;6Y[10XModuleGeneratorsOverOriginalMonoid[110X                                 see
        [2XNmzModuleGeneratorsOverOriginalMonoid[102X ([14X2.3-26[114X),[133X
  
  [30X    [33X[0;6Y[10XSublattice[110X  computes  the  efficient  sublattice  and  returns  a bool
        signaling   whether   the  computation  was  successful.  Actual  data
        connected  to  it  can  be  accessed by [2XNmzRank[102X ([14X2.3-30[114X), [2XNmzEquations[102X
        ([14X2.3-7[114X), [2XNmzCongruences[102X ([14X2.3-3[114X), and [2XNmzBasisChange[102X ([14X2.3-51[114X).[133X
  
  [33X[0;0YAdditionally  also  the  following compute options are accepted as property.
  They  modify  what  and  how  should  be  computed,  and return True after a
  successful computation.[133X
  
  [30X    [33X[0;6Y[10XApproximate[110X approximate the rational polytope by an integral polytope,
        currently only useful in combination with [10XDeg1Elements[110X.[133X
  
  [30X    [33X[0;6Y[10XBottomDecomposition[110X  use the best possible triangulation (with respect
        to the sum of determinants) using the given generators.[133X
  
  [30X    [33X[0;6Y[10XDefaultMode[110X  try  to  compute  what  is  possible  and do not throw an
        exception when something cannot be computed.[133X
  
  [30X    [33X[0;6Y[10XDualMode[110Xactivates  the  dual  algorithm  for  the  computation  of the
        Hilbert  basis  and  degree  1 elements. Includes [10XHilbertBasis[110X, unless
        [10XDeg1Elements[110X   is   set.  Often  a  good  choice  if  you  start  from
        constraints.[133X
  
  [30X    [33X[0;6Y[10XKeepOrder[110X     forbids    to    reorder    the    generators.    Blocks
        [10XBottomDecomposition[110X.[133X
  
  [33X[0;0YAll  the  properties above can be given to [2XNmzCompute[102X ([14X2.2-5[114X). There you can
  combine  different properties, e.g. give some properties that you would like
  to know and add some compute options.[133X
  
  [33X[0;0YSee the Normaliz manual for a detailed description.[133X
  
  [1X2.2-7 NmzPrintConeProperties[101X
  
  [29X[2XNmzPrintConeProperties[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YPrint  an  overview  of  all  known properties of the given cone, as well as
  their values.[133X
  
  
  [1X2.3 [33X[0;0YCone properties[133X[101X
  
  [1X2.3-1 NmzAffineDim[101X
  
  [29X[2XNmzAffineDim[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe affine dimension[133X
  
  [33X[0;0YThe  affine  dimension  of the polyhedron in inhomogeneous computations. Its
  computation is triggered if necessary.[133X
  
  [33X[0;0YThis   is   an   alias   for   [10XNmzConeProperty(  cone,  "AffineDim"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-2 NmzClassGroup[101X
  
  [29X[2XNmzClassGroup[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe class group in a special format[133X
  
  [33X[0;0YA  normal  affine  monoid  [23XM[123X  has  a well-defined divisor class group. It is
  naturally  isomorphic  to the divisor class group of [23XK[M][123X where [23XK[123X 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 [22Xn,m[122X.
  Such  two  entries  represent a free cyclic summand [22X(Z/nZ)^m[122X. Not allowed in
  inhomogeneous computations.[133X
  
  [33X[0;0YThis   is   an   alias   for  [10XNmzConeProperty(  cone,  "ClassGroup"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-3 NmzCongruences[101X
  
  [29X[2XNmzCongruences[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows represent the congruences[133X
  
  [33X[0;0YThe  equations,  congruences  and  support hyperplanes together describe the
  lattice points of the cone.[133X
  
  [33X[0;0YThis is part of the cone property [21XSublattice[121X.[133X
  
  [1X2.3-4 NmzDeg1Elements[101X
  
  [29X[2XNmzDeg1Elements[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the degree 1 elements[133X
  
  [33X[0;0YRequires   the   presence   of  a  grading.  Not  allowed  in  inhomogeneous
  computations.[133X
  
  [33X[0;0YThis   is   an  alias  for  [10XNmzConeProperty(  cone,  "Deg1Elements"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-5 NmzDehomogenization[101X
  
  [29X[2XNmzDehomogenization[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe dehomgenization vector[133X
  
  [33X[0;0YOnly for inhomogeneous computations.[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty(  cone,  "Dehomogenization" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-6 NmzEmbeddingDimension[101X
  
  [29X[2XNmzEmbeddingDimension[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe embedding dimension of the cone[133X
  
  [33X[0;0YThe  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.[133X
  
  [1X2.3-7 NmzEquations[101X
  
  [29X[2XNmzEquations[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows represent the equations[133X
  
  [33X[0;0YThe equations cut out the linear space generated by the cone. The equations,
  congruences  and support hyperplanes together describe the lattice points of
  the cone.[133X
  
  [1X2.3-8 NmzExcludedFaces[101X
  
  [29X[2XNmzExcludedFaces[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows represent the excluded faces[133X
  
  [33X[0;0YThis   is  an  alias  for  [10XNmzConeProperty(  cone,  "ExcludedFaces"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-9 NmzExtremeRays[101X
  
  [29X[2XNmzExtremeRays[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the extreme rays[133X
  
  [33X[0;0YThis   is   an   alias  for  [10XNmzConeProperty(  cone,  "ExtremeRays"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-10 NmzGenerators[101X
  
  [29X[2XNmzGenerators[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the generators of [3Xcone[103X[133X
  
  [33X[0;0YThis   is   an   alias   for  [10XNmzConeProperty(  cone,  "Generators"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-11 NmzGeneratorOfInterior[101X
  
  [29X[2XNmzGeneratorOfInterior[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya vector representing the generator of the interior of [3Xcone[103X[133X
  
  [33X[0;0YIf  [3Xcone[103X  is Gorenstein, this function returns the generator of the interior
  of [3Xcone[103X. If [3Xcone[103X is not Gorenstein, an error is raised.[133X
  
  [1X2.3-12 NmzGrading[101X
  
  [29X[2XNmzGrading[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe grading vector[133X
  
  [33X[0;0YThis is an alias for [10XNmzConeProperty( cone, "Grading" );[110X see [2XNmzConeProperty[102X
  ([14X2.2-6[114X).[133X
  
  [1X2.3-13 NmzHilbertBasis[101X
  
  [29X[2XNmzHilbertBasis[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the Hilbert basis elements[133X
  
  [33X[0;0YThis   is   an  alias  for  [10XNmzConeProperty(  cone,  "HilbertBasis"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-14 NmzHilbertQuasiPolynomial[101X
  
  [29X[2XNmzHilbertQuasiPolynomial[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe Hilbert function as a quasipolynomial[133X
  
  [33X[0;0YThe  Hilbert  function counts the lattice points degreewise. The result is a
  quasipolynomial  [22XQ[122X,  that is, a polynomial with periodic coefficients. It is
  given  as list of polynomials [22XP_0, ..., P_(p-1)[122X such that [22XQ(i) = P_(i mod p)
  (i)[122X.[133X
  
  [33X[0;0YThis  is an alias for [10XNmzConeProperty( cone, "HilbertQuasiPolynomial" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-15 NmzHilbertSeries[101X
  
  [29X[2XNmzHilbertSeries[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe Hilbert series as rational function[133X
  
  [33X[0;0YThe  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 [22X[k_i, l_i][122X. Such a pair represents the factor [22X(1-t^k_i)^l_i[122X.[133X
  
  [33X[0;0YThis   is  an  alias  for  [10XNmzConeProperty(  cone,  "HilbertSeries"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-16 NmzInclusionExclusionData[101X
  
  [29X[2XNmzInclusionExclusionData[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Yinclusion-exclusion data[133X
  
  [33X[0;0YList 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  [2XNmzTriangulation[102X ([14X2.3-34[114X)) and the
  multiplicity  with  which  it  was  considered. Only available with excluded
  faces or strict constraints as input.[133X
  
  [33X[0;0YThis  is an alias for [10XNmzConeProperty( cone, "InclusionExclusionData" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-17 NmzIsDeg1ExtremeRays[101X
  
  [29X[2XNmzIsDeg1ExtremeRays[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if all extreme rays have degree 1; [9Xfalse[109X otherwise[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty(  cone, "IsDeg1ExtremeRays" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-18 NmzIsDeg1HilbertBasis[101X
  
  [29X[2XNmzIsDeg1HilbertBasis[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if all Hilbert basis elements have degree 1; [9Xfalse[109X otherwise[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty( cone, "IsDeg1HilbertBasis" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-19 NmzIsGorenstein[101X
  
  [29X[2XNmzIsGorenstein[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ywhether the cone is Gorenstein[133X
  
  [33X[0;0YReturns true if [3Xcone[103X is Gorenstein, false otherwise.[133X
  
  [1X2.3-20 NmzIsInhomogeneous[101X
  
  [29X[2XNmzIsInhomogeneous[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ywhether the cone is inhomogeneous[133X
  
  [33X[0;0YThis value is always known directly after the creation of the cone.[133X
  
  [1X2.3-21 NmzIsIntegrallyClosed[101X
  
  [29X[2XNmzIsIntegrallyClosed[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if the cone is integrally closed; [9Xfalse[109X otherwise[133X
  
  [33X[0;0YIt  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.[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty( cone, "IsIntegrallyClosed" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-22 NmzIsPointed[101X
  
  [29X[2XNmzIsPointed[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if the cone is pointed; [9Xfalse[109X otherwise[133X
  
  [33X[0;0YThis   is   an   alias   for   [10XNmzConeProperty(  cone,  "IsPointed"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-23 NmzIsReesPrimary[101X
  
  [29X[2XNmzIsReesPrimary[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if is the monomial ideal is primary to the irrelevant maximal
            ideal, [9Xfalse[109X otherwise[133X
  
  [33X[0;0YOnly used with the input type [10Xrees_algebra[110X.[133X
  
  [33X[0;0YThis   is  an  alias  for  [10XNmzConeProperty(  cone,  "IsReesPrimary"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-24 NmzMaximalSubspace[101X
  
  [29X[2XNmzMaximalSubspace[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows generate the maximale linear subspace[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty(  cone,  "MaximalSubspace"  );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-25 NmzModuleGenerators[101X
  
  [29X[2XNmzModuleGenerators[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the module generators[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty(  cone,  "ModuleGenerators" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-26 NmzModuleGeneratorsOverOriginalMonoid[101X
  
  [29X[2XNmzModuleGeneratorsOverOriginalMonoid[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  matrix  whose  rows are the module generators over the original
            monoid[133X
  
  [33X[0;0YA  minimal  system  of  generators of the integral closure over the original
  monoid. Requires the existence of original monoid generators. Not allowed in
  inhomogeneous computations.[133X
  
  [33X[0;0YThis       is       an       alias      for      [10XNmzConeProperty(      cone,
  "ModuleGeneratorsOverOriginalMonoid" );[110X see [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-27 NmzModuleRank[101X
  
  [29X[2XNmzModuleRank[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe  rank  of  the module of lattice points in the polyhedron as a
            module over the recession monoid[133X
  
  [33X[0;0YOnly for inhomogeneous computations.[133X
  
  [33X[0;0YThis   is   an   alias   for  [10XNmzConeProperty(  cone,  "ModuleRank"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-28 NmzMultiplicity[101X
  
  [29X[2XNmzMultiplicity[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis   is   an  alias  for  [10XNmzConeProperty(  cone,  "Multiplicity"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-29 NmzOriginalMonoidGenerators[101X
  
  [29X[2XNmzOriginalMonoidGenerators[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the original monoid generators[133X
  
  [33X[0;0YThis  is  an  alias for [10XNmzConeProperty( cone, "OriginalMonoidGenerators" );[110X
  see [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-30 NmzRank[101X
  
  [29X[2XNmzRank[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe rank of the cone[133X
  
  [33X[0;0YThis value is the rank of the lattice generated by the lattice points of the
  cone.[133X
  
  [33X[0;0YThis is part of the cone property [21XSublattice[121X.[133X
  
  [1X2.3-31 NmzRecessionRank[101X
  
  [29X[2XNmzRecessionRank[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe rank of the recession cone[133X
  
  [33X[0;0YOnly for inhomogeneous computations.[133X
  
  [33X[0;0YThis   is  an  alias  for  [10XNmzConeProperty(  cone,  "RecessionRank"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-32 NmzReesPrimaryMultiplicity[101X
  
  [29X[2XNmzReesPrimaryMultiplicity[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0Ythe  multiplicity of a monomial ideal, provided it is primary to the maximal
  ideal  generated  by  the  indeterminates.  Used  only  with  the input type
  [10Xrees_algebra[110X.[133X
  
  [33X[0;0YThis is an alias for [10XNmzConeProperty( cone, "ReesPrimaryMultiplicity" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-33 NmzSupportHyperplanes[101X
  
  [29X[2XNmzSupportHyperplanes[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows represent the support hyperplanes[133X
  
  [33X[0;0YThe equations cut out the linear space generated by the cone. The equations,
  congruences  and support hyperplanes together describe the lattice points of
  the cone.[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty( cone, "SupportHyperplanes" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-34 NmzTriangulation[101X
  
  [29X[2XNmzTriangulation[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe triangulation[133X
  
  [33X[0;0YThis  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 [23Xa_1,\dots,a_n[123X referring to the
  [2XNmzGenerators[102X  ([14X2.3-10[114X)  (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.[133X
  
  [33X[0;0YThis   is  an  alias  for  [10XNmzConeProperty(  cone,  "Triangulation"  );[110X  see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-35 NmzTriangulationDetSum[101X
  
  [29X[2XNmzTriangulationDetSum[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ysum  of  the absolute values of the determinants of the simplicial
            cones in the used triangulation[133X
  
  [33X[0;0YThis  is  an  alias  for [10XNmzConeProperty( cone, "TriangulationDetSum" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-36 NmzTriangulationSize[101X
  
  [29X[2XNmzTriangulationSize[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe number of simplicial cones in the used triangulation[133X
  
  [33X[0;0YThis  is  an  alias  for  [10XNmzConeProperty(  cone, "TriangulationSize" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-37 NmzVerticesFloat[101X
  
  [29X[2XNmzVerticesFloat[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  matrix  whose rows are the vertices of the polyhedron [3Xcone[103X with
            float coordinates[133X
  
  [33X[0;0YThe  rows  of  this matrix represent the vertices of [3Xcone[103X, printed as floats
  for  better  readability.  The result might be inexact, and should therefore
  not be used for computations.[133X
  
  [1X2.3-38 NmzVerticesOfPolyhedron[101X
  
  [29X[2XNmzVerticesOfPolyhedron[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya matrix whose rows are the vertices of the polyhedron[133X
  
  [33X[0;0YThis  is  an  alias for [10XNmzConeProperty( cone, "VerticesOfPolyhedron" );[110X see
  [2XNmzConeProperty[102X ([14X2.2-6[114X).[133X
  
  [1X2.3-39 NmzConeDecomposition[101X
  
  [29X[2XNmzConeDecomposition[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "ConeDecomposition" );[133X
  
  [1X2.3-40 NmzEmbeddingDim[101X
  
  [29X[2XNmzEmbeddingDim[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "EmbeddingDim" );[133X
  
  [1X2.3-41 NmzExternalIndex[101X
  
  [29X[2XNmzExternalIndex[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "ExternalIndex" );[133X
  
  [1X2.3-42 NmzGradingDenom[101X
  
  [29X[2XNmzGradingDenom[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "GradingDenom" );[133X
  
  [1X2.3-43 NmzIntegerHull[101X
  
  [29X[2XNmzIntegerHull[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "IntegerHull" );[133X
  
  [1X2.3-44 NmzInternalIndex[101X
  
  [29X[2XNmzInternalIndex[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "InternalIndex" );[133X
  
  [1X2.3-45 NmzStanleyDec[101X
  
  [29X[2XNmzStanleyDec[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "StanleyDec" );[133X
  
  [1X2.3-46 NmzSublattice[101X
  
  [29X[2XNmzSublattice[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "Sublattice" );[133X
  
  [1X2.3-47 NmzUnitGroupIndex[101X
  
  [29X[2XNmzUnitGroupIndex[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "UnitGroupIndex" );[133X
  
  [1X2.3-48 NmzWeightedEhrhartQuasiPolynomial[101X
  
  [29X[2XNmzWeightedEhrhartQuasiPolynomial[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis       is       an       alias      for      NmzConeProperty(      cone,
  "NmzWeightedEhrhartQuasiPolynomial" );[133X
  
  [1X2.3-49 NmzWeightedEhrhartSeries[101X
  
  [29X[2XNmzWeightedEhrhartSeries[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartSeries" );[133X
  
  [1X2.3-50 NmzWitnessNotIntegrallyClosed[101X
  
  [29X[2XNmzWitnessNotIntegrallyClosed[102X( [3Xcone[103X ) [32X function
  
  [33X[0;0YThis is an alias for NmzConeProperty( cone, "WitnessNotIntegrallyClosed" );[133X
  
  [1X2.3-51 NmzBasisChange[101X
  
  [29X[2XNmzBasisChange[102X( [3Xcone[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record describing the basis change[133X
  
  [33X[0;0YThe  result  record  [10Xr[110X  has three components: [10Xr.Embedding[110X, [10Xr.Projection[110X, and
  [10Xr.Annihilator[110X,  where the embedding [10XA[110X and the projection [10XB[110X are matrices, and
  the  annihilator  [10Xc[110X  is  an  integer.  They  represent  the mapping into the
  effective  lattice  [22XZ^n  -> Z^r, u ↦ (uB)/c[122X and the inverse operation [22XZ^r ->
  Z^n, v ↦ vA[122X.[133X
  
  [33X[0;0YThis is part of the cone property [21XSublattice[121X.[133X
  
