aperyConeEquations -- Inequalities defining the Kunz cones

Usage:

M = aperyConeEquations m

M = aperyConeEquations sgrp

M = muConeEquations m

M = muConeEquations sgrp
Inputs:
- m, an integer, multiplicity
- sgrp, a list, generators of a semigroup
Outputs:
- M, a matrix, m-1 x d matrix whose columns represent inequalities defining the Kunz cone

Description

Let S be the numerical semigroup defined by a list sgrp, have multiplicity m and apery set a_1,\dots,a_(m-1) and set mu_i = (a_i -i)//m. The homogeneous Kunz cone of semigroups of multiplity m is the convex polyhedral cone defined by the inequalities of the form

a_i + a_j - a_(i+j) \geq 0.

where 1\leq i,j\leq m-1 and i+j\neq m is interpreted mod m.

On the other hand, the inHomogeneous Kunz cone is given by the equations

mu_i + m_j - mu_(i+j) \geq 0 if i+j<m mu_i + m_j - mu_(i+j) -1 \geq 0 if i+j> m

The output of the function is the pair (E,c), where, if L is a semigroup, then

matrix {Apery L} * E

is a non-negative matrix with entries divisible by m, and

matrix {mu L} *E - c

is a non-negative matrix. The columns of E and c are indexed by the lexicographically ordered pairs {i,j} with 1<= i<=j<=m-1 and i+j != m.

The function aperyConeEquations m returns an m-1 x d matrix of ZZ whose columns are the coefficients of the left hand sides of these inequalities. The function aperyConeEquations sgrp does the same, with additional columns representing the additional inequalities of this type that are satisfied by the Apery set apery(sgrp). For m = 3, the semigroup {3,4,5} is interior (and thus satisfies no further equations), while the semigroups {3,4} and {3,5} are on the two extremal rays of the cone.

i1 : aperyConeEquations 3

o1 = | 2  -1 |
     | -1 2  |

              2       2
o1 : Matrix ZZ  <-- ZZ

i2 : aperyConeEquations {3,4,5}

o2 = | 2  -1 |
     | -1 2  |

              2       2
o2 : Matrix ZZ  <-- ZZ

i3 : aperyConeEquations {3,4}

o3 = | 2  -1 -2 |
     | -1 2  1  |

              2       3
o3 : Matrix ZZ  <-- ZZ

i4 : aperyConeEquations {3,5}

o4 = | 2  -1 1  |
     | -1 2  -2 |

              2       3
o4 : Matrix ZZ  <-- ZZ

i5 : allSemigroups 3

o5 = (| 3 3 |, | 4  5 |)
      | 3 6 |  | 4  8 |
      | 6 3 |  | 7  5 |
               | 10 5 |

o5 : Sequence

The inhomogeneous Kunz cone does the same, but for the numbers mu_i instead of a_i. Thus when i+j > m the inequality mu_i+mu_j-mu_(i+j) \geq 0 is replaced by the inequality

mu_i+mu_j - mu_(i+j) -1.

The function muConeEquations m, returns the matrix aperyConeEquations m with one more row, where the last row represents the constant terms of this inquality:

i6 : eq=muConeEquations 3

o6 = | 2  -1 |
     | -1 2  |
     | 0  1  |

              3       2
o6 : Matrix ZZ  <-- ZZ

i7 : muConeEquations {3,4,5}

o7 = | 2  -1 |
     | -1 2  |
     | 0  1  |

              3       2
o7 : Matrix ZZ  <-- ZZ

i8 : muConeEquations {3,4}

o8 = | 2  -1 -2 |
     | -1 2  1  |
     | 0  1  0  |

              3       3
o8 : Matrix ZZ  <-- ZZ

i9 : muConeEquations {3,5}

o9 = | 2  -1 1  |
     | -1 2  -2 |
     | 0  1  0  |

              3       3
o9 : Matrix ZZ  <-- ZZ

i10 : (H,M)=allSemigroups 3

o10 = (| 3 3 |, | 4  5 |)
       | 3 6 |  | 4  8 |
       | 6 3 |  | 7  5 |
                | 10 5 |

o10 : Sequence

i11 : (H,M)=allSemigroups 4

o11 = (| 4  0 4  |, | 5  6  7  |)
       | 4  4 4  |  | 5  6  11 |
       | 4  4 8  |  | 5  10 7  |
       | 4  8 4  |  | 5  10 11 |
       | 4  8 8  |  | 5  10 15 |
       | 4  8 12 |  | 9  6  7  |
       | 8  4 4  |  | 9  10 7  |
       | 8  8 4  |  | 9  14 7  |
       | 12 8 4  |  | 13 6  7  |
                    | 13 10 7  |
                    | 13 14 7  |
                    | 17 10 7  |
                    | 17 14 7  |
                    | 21 14 7  |

o11 : Sequence

i12 : M1=(M|matrix apply(rank target M,i->{1}))

o12 = | 5  6  7  1 |
      | 5  6  11 1 |
      | 5  10 7  1 |
      | 5  10 11 1 |
      | 5  10 15 1 |
      | 9  6  7  1 |
      | 9  10 7  1 |
      | 9  14 7  1 |
      | 13 6  7  1 |
      | 13 10 7  1 |
      | 13 14 7  1 |
      | 17 10 7  1 |
      | 17 14 7  1 |
      | 21 14 7  1 |

               14       4
o12 : Matrix ZZ   <-- ZZ

i13 : eqInh=muConeEquations 4

o13 = | 2  1  -1 0  |
      | -1 1  1  -1 |
      | 0  -1 1  2  |
      | 0  0  1  1  |

               4       4
o13 : Matrix ZZ  <-- ZZ

i14 : eqh=aperyConeEquations 4

o14 = | 2  1  -1 0  |
      | -1 1  1  -1 |
      | 0  -1 1  2  |

               3       4
o14 : Matrix ZZ  <-- ZZ

i15 : M1*eqInh

o15 = | 4  4  9  9  |
      | 4  0  13 17 |
      | 0  8  13 5  |
      | 0  4  17 13 |
      | 0  0  21 21 |
      | 12 8  5  9  |
      | 8  12 9  5  |
      | 4  16 13 1  |
      | 20 12 1  9  |
      | 16 16 5  5  |
      | 12 20 9  1  |
      | 24 20 1  5  |
      | 20 24 5  1  |
      | 28 28 1  1  |

               14       4
o15 : Matrix ZZ   <-- ZZ

i16 : H*eqh

o16 = | 8  0  0  8  |
      | 4  4  4  4  |
      | 4  0  8  12 |
      | 0  8  8  0  |
      | 0  4  12 8  |
      | 0  0  16 16 |
      | 12 8  0  4  |
      | 8  12 4  0  |
      | 16 16 0  0  |

               9       4
o16 : Matrix ZZ  <-- ZZ

All entries of M1*eqInh and H*eqh are non-negative as desired.

References

Kunz, Ernst: Klassification numerische Halbgruppen

Caveat

Ways to use aperyConeEquations:

aperyConeEquations(List)
aperyConeEquations(ZZ)

For the programmer

The object aperyConeEquations is a method function.