randomSemigroup -- Random semingroup on a given face of the Kunz cone

Usage:

L' = randomSemigroup(L,b)

L' = randomSemigroup(m,b)
Inputs:
- L, a list, of generators of a semigroup
- b, an integer, bound for the random numbers to be used
- m, an integer, multiplicity
Outputs:
- L', a list, Apery set of a semigroup

Description

Find a random semigroup within a bounded portiion of the open face of the Kunz cone containing L (or of the interior of the Kunz cone, if a multiplicity m is specified instead of a list.

After (H,M) = allSemigroups L, the Apery set of L' is obtained by prepending m to the sum of a random linear combination of the rows of H (using random integers 0..b-1) and a random row of M.

Since L' is on the same face as L, it shares many homological properties, such as the total betti numbers of the resolution of its semigroup ideal.

i1 : L = {6,9,13,16}

o1 = {6, 9, 13, 16}

o1 : List

i2 : L' = randomSemigroup({6,9,13,16}, 5)

o2 = {6, 121, 242, 255, 226, 347}

o2 : List

i3 : mingens L'

o3 = {6, 121, 226, 255}

o3 : List

i4 : F = res semigroupIdeal L;

i5 : F'= res semigroupIdeal L';

i6 : apply(1+length F, i-> rank F_i)

o6 = {1, 6, 8, 3}

o6 : List

i7 : apply(1+length F', i-> rank F'_i)

o7 = {1, 6, 8, 3}

o7 : List

Caveat

The list L' is an Apery set, so may not be a minimal set of semigroup generators.

Ways to use randomSemigroup:

randomSemigroup(List,ZZ)
randomSemigroup(ZZ,ZZ)

For the programmer

The object randomSemigroup is a method function.

The source of this document is in NumericalSemigroups.m2:2299:0.

randomSemigroup -- Random semingroup on a given face of the Kunz cone

Description

Caveat

See also

Ways to use randomSemigroup:

For the programmer