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# ehrhartRing(List,allComputations=>...) -- Ehrhart ring

- Usage:
    `ehrhartRing L`
- Inputs:
  - a list, the monomials corresponding to the vertices of a lattice polytope
- Outputs:
  - a monomial subalgebra, generated by the lattice points of the polytope
  - a monomial subalgebra, the Ehrhart ring of the polytope

## Description

The exponent vectors of the given monomials are considered as vertices of a lattice polytope P. The Ehrhart ring of a (lattice) polytope P is the monoid algebra defined by the monoid of lattice points in the cone over the polytope P; see Bruns and Gubeladze, Polytopes, Rings, and K-theory, Springer 2009, pp. 228, 229. The function returns two monomial subalgebras, the first has as generators the monomials representing the lattice points of the polytope, the second is the Ehrhart ring. Since these are defined in a polynomial ring with an additional variable, the function creates a new polynomial ring with an additional variable. If the option allComputations is set to true, all data that has been computed by Normaliz is stored in a RationalCone in the CacheTable of the monomial subalgebras returned (it is the same cone for both subalgebras).

    i1 : R=ZZ/37[x,y];

    i2 : L={x^3, x^2*y, y^3, x*y^2};

    i3 : (A, B) = ehrhartRing(L, allComputations => true);

    i4 : A.cache#"cone"

    o4 = RationalCone{"cgr" => | 0 |                                          }
                               | 4 |
                      "equ" => | 1 1 -3 |
                      "gen" => | 0 3 1 |
                               | 1 2 1 |
                               | 2 1 1 |
                               | 3 0 1 |
                      "inv" => HashTable{"" => (1, 1)                        }
                                         "class group" => (0, 3)
                                         "degree 1 elements" => 4
                                         "dim max subspace" => 0
                                         "embedding dim" => 3
                                         "external index" => 1
                                         "graded" => true
                                         "grading denom" => 1
                                         "grading" => (0, 0, 1)
                                         "hilbert basis elements" => 4
                                         "hilbert quasipolynomial denom" => 1
                                         "hilbert series denom" => (1, 1)
                                         "hilbert series num" => (1, 2)
                                         "inhomogeneous" => false
                                         "integrally closed" => true
                                         "internal index" => 1
                                         "multiplicity denom" => 1
                                         "multiplicity" => 3
                                         "number extreme rays" => 2
                                         "number support hyperplanes" => 2
                                         "rank" => 2
                                         "size triangulation" => 3
                                         "sum dets" => 3
                      "sup" => | -1 0 3 |
                               | 1  0 0 |

    o4 : RationalCone

    i5 : B.cache#"cone"

    o5 = RationalCone{"cgr" => | 0 |                                          }
                               | 4 |
                      "equ" => | 1 1 -3 |
                      "gen" => | 0 3 1 |
                               | 1 2 1 |
                               | 2 1 1 |
                               | 3 0 1 |
                      "inv" => HashTable{"" => (1, 1)                        }
                                         "class group" => (0, 3)
                                         "degree 1 elements" => 4
                                         "dim max subspace" => 0
                                         "embedding dim" => 3
                                         "external index" => 1
                                         "graded" => true
                                         "grading denom" => 1
                                         "grading" => (0, 0, 1)
                                         "hilbert basis elements" => 4
                                         "hilbert quasipolynomial denom" => 1
                                         "hilbert series denom" => (1, 1)
                                         "hilbert series num" => (1, 2)
                                         "inhomogeneous" => false
                                         "integrally closed" => true
                                         "internal index" => 1
                                         "multiplicity denom" => 1
                                         "multiplicity" => 3
                                         "number extreme rays" => 2
                                         "number support hyperplanes" => 2
                                         "rank" => 2
                                         "size triangulation" => 3
                                         "sum dets" => 3
                      "sup" => | -1 0 3 |
                               | 1  0 0 |

    o5 : RationalCone

## Functions with optional argument named allComputations:

- diagInvariants(...,allComputations=>...) (missing documentation)
- ehrhartRing(...,allComputations=>...) (missing documentation)
- finiteDiagInvariants(...,allComputations=>...) (missing documentation)
- intclMonIdeal(...,allComputations=>...) (missing documentation)
- intclToricRing(...,allComputations=>...) (missing documentation)
- intersectionValRingIdeals(...,allComputations=>...) (missing documentation)
- intersectionValRings(...,allComputations=>...) (missing documentation)
- normaliz(...,allComputations=>...) (missing documentation)
- normalToricRing(...,allComputations=>...) (missing documentation)
- torusInvariants(...,allComputations=>...) (missing documentation)

## Further information

- Default value: false
- Method: ehrhartRing(List) -- Ehrhart ring
- Option key: allComputations

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Normaliz.m2:1744:0.