kunzRing -- artinian reduction of a semigroup ring

Usage:

R = kunzRing L
Inputs:
- L, a list, of ZZ
Optional inputs:
- BaseField
- MinimalGenerators
- VariableName
Outputs:
- R, a ring, semigroup ring mod multiplicity element

Description

Returns the semigroup ring modulo the element of least degree. The kunzRing shares many properties with the semigroup ring; see semigroupRing for explanations of the options.

i1 : semigroupRing {3,5}

       ZZ
     -----[x , x ]
     10007  0   2
o1 = -------------
         5    3
        x  - x
         0    2

o1 : QuotientRing

i2 : kunzRing {3,5}

       ZZ
     -----[x ]
     10007  2
o2 = ---------
          3
         x
          2

o2 : QuotientRing

i3 : kunzRing ({3,5,6}, "BaseField" => ZZ/32003, "VariableName" => y, "MinimalGenerators" => false)

       ZZ
     -----[y , y   ]
     32003  2   1,0
o3 = ---------------
                3
        (y   , y )
          1,0   2

o3 : QuotientRing

The Kunz ring is an invariant of the face of the Kunz cone which contains L. For all L in the interior of the corresponding face have isomorphic Kunz rings.

i4 : L=semigroup {4,6,7}

o4 = {0, 4, 6, 7, 8, 10, 11, 12, 13}

o4 : List

i5 : (H,M)=allSemigroups {4,6,7}

o5 = (| 4  0 4 |, | 13 6  7 |)
      | 8  4 4 |  | 17 10 7 |
      | 12 8 4 |  | 21 14 7 |

o5 : Sequence

i6 : L1={4}|flatten (entries(M^{2}+3*H^{1}))

o6 = {4, 45, 26, 19}

o6 : List

i7 : #gaps L1, socle L1

o7 = (21, {45})

o7 : Sequence

i8 : kunzRing {4,6,7}

       ZZ
     -----[x ..x ]
     10007  2   3
o8 = -------------
          2   2
        (x , x )
          2   3

o8 : QuotientRing

i9 : kunzRing L1

       ZZ
     -----[x , x ]
     10007  3   2
o9 = -------------
          2   2
        (x , x )
          3   2

o9 : QuotientRing

Ways to use kunzRing:

kunzRing(List)

For the programmer

The object kunzRing is a method function with options.

The source of this document is in NumericalSemigroups.m2:2698:0.