kunzRing -- artinian reduction of a semigroup ring

Synopsis

Usage:

R = kunzRing L
Inputs:
- L, a list, of ZZ
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.

i1 : semigroupRing {3,5}

      ZZ
     ---[x , x ]
     101  0   2
o1 = -----------
        5    3
       x  - x
        0    2

o1 : QuotientRing

i2 : kunzRing {3,5}

      ZZ
     ---[x ]
     101  2
o2 = -------
         3
        x
         2

o2 : 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.

i3 : L=semigroup {4,6,7}

o3 = {0, 4, 6, 7, 8, 10, 11, 12, 13}

o3 : List

i4 : (H,M)=allSemigroups {4,6,7}

o4 = (| 4  0 4 |, | 13 6  7 |)
      | 8  4 4 |  | 17 10 7 |
      | 12 8 4 |  | 21 14 7 |

o4 : Sequence

i5 : L1={4}|flatten (entries(M^{2}+3*H^{1}))

o5 = {4, 45, 26, 19}

o5 : List

i6 : #gaps L1, socle L1

o6 = (21, {45})

o6 : Sequence

i7 : kunzRing {4,6,7}

      ZZ
     ---[x ..x ]
     101  2   3
o7 = -----------
         2   2
       (x , x )
         2   3

o7 : QuotientRing

i8 : kunzRing L1

      ZZ
     ---[x , x ]
     101  3   2
o8 = -----------
         2   2
       (x , x )
         3   2

o8 : QuotientRing

Ways to use kunzRing:

kunzRing(List)

For the programmer

The object kunzRing is a method function.