facetRays -- computes the rays spanning the face in which a semigroup lies

Usage:

R = facetRays L
Inputs:
- L, a list, of generators of a semigroup
Outputs:
- R, a matrix, of integers; the columns are the rays of the face on which semigroup L lies

Description

Note that this function computes the rays of the closed face *not facet* on which L lies. Uses the Fourier-Motzkin algorithm to go from the aperyConeEquations satisfied by the semigroup to the rays. For example, in multiplicity 3, the cone has two rays, occupied by the semigroups semigroup{3,4} and semigroup{3,5}, with semigroup{3,4,5} in the interior. The rays are given in reduced form (a vector of positive integers with gcd 1), and appear as the columns of the output matrix.

i1 : aperySet{3,4}

o1 = {4, 8}

o1 : List

i2 : facetRays{3,4}

o2 = | 1 |
     | 2 |

              2       1
o2 : Matrix ZZ  <-- ZZ

i3 : facetRays{3,5}

o3 = | 2 |
     | 1 |

              2       1
o3 : Matrix ZZ  <-- ZZ

i4 : facetRays{3,4,5}

o4 = | 1 2 |
     | 2 1 |

              2       2
o4 : Matrix ZZ  <-- ZZ

On the facet with the buchweitz example there are two facet rays:

i5 : F = facetRays buchweitz 0

o5 = | 1  1 |
     | 2  1 |
     | 3  1 |
     | 4  1 |
     | 5  1 |
     | 6  2 |
     | 7  1 |
     | 8  2 |
     | 9  1 |
     | 10 1 |
     | 11 2 |
     | 12 2 |

              12       2
o5 : Matrix ZZ   <-- ZZ

The second column is mu buchweitz 0, the mu vector of the Buchweitz example. Adding multiples of it to the Weierstrass semigroups ordinary point of genus 12, we eventually reach a semigroup that fails the Buchweitz test to be a Weierstrass semigroup:

i6 : b =flatten( {0}| entries(F)_0 ) -- edited for minimal change STILL TO BE FIXED

o6 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

o6 : List

i7 : L = toList (13..25)

o7 = {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}

o7 : List

i8 : #L,#b

o8 = (13, 13)

o8 : Sequence

We conjecture that the same phenomen for any semigroup L0 of multiplicity 13 in place of L. Here is a "random" example:

Ways to use facetRays:

facetRays(List)

For the programmer

The object facetRays is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/NumericalSemigroups.m2:2149:0.

facetRays -- computes the rays spanning the face in which a semigroup lies

Description

See also

Ways to use facetRays:

For the programmer