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semigroupRing -- forms the semigroup ring over "BaseField"

Synopsis

Description

If the basering is kk, the semigroup ring is A = kk[x^S] where x^S denotes the set of monomials in variables x_i with exponent vectors in S, and kk is the field that is the value of the option "BaseField" (ZZ/101 by default).

If m is the multiplicity of S, the semigroup ring depends up to an isomorphism that may change the degrees, only on the face of the Kunz cone in which the semigroup lies.

Semigroup rings are interesting as examples, and arise as Weierstrass semigroups of points on algebraic curves: if p in C is such a point, then the Weierstrass semigroup of C at p is the set of pole orders of rational functions with poles only at p, and the semigroupRing is the associated graded ring of the filtered ring $\bigcup_{n >= 0} H^0(O_C(np))$. For non-Weierstrass points the semigroup is 0,g+1,g+2.., and there are finitely many "Weierstrass" point p whose semigroup has weight >=2.

For example if C is a smooth plane quartic, then at each ordinary flex point, the semigroup is 0,3,5,6,7,..

i1 : semigroupRing {3,4,5}

                 ZZ
                ---[x ..x ]
                101  0   2
o1 = ---------------------------------
       2          3          2      2
     (x  - x x , x  - x x , x x  - x )
       1    0 2   0    1 2   0 1    2

o1 : QuotientRing
 
i2 : weight {4,5,6,7}, gaps {4,5,6,7}

o2 = (0, {1, 2, 3})

o2 : Sequence
 
i3 : semigroupRing {4,5,6,7}

                                    ZZ
                                   ---[x ..x ]
                                   101  0   3
o3 = ----------------------------------------------------------------------
       2                       2          3          2            2      2
     (x  - x x , x x  - x x , x  - x x , x  - x x , x x  - x x , x x  - x )
       1    0 2   1 2    0 3   2    1 3   0    1 3   0 1    2 3   0 2    3

o3 : QuotientRing
 
i4 : weight {3,5,7}, gaps {3,5,7}

o4 = (1, {1, 2, 4})

o4 : Sequence
 
i5 : semigroupRing {3,5,7}

               ZZ
              ---[x , x , x ]
              101  0   2   1
o5 = ---------------------------------
       2          4          3      2
     (x  - x x , x  - x x , x x  - x )
       2    0 1   0    2 1   0 2    1

o5 : QuotientRing
 
i6 : weight {3,4}, gaps {3,4}

o6 = (2, {1, 2, 5})

o6 : Sequence
 
i7 : semigroupRing({3,4}, "BaseField" => QQ)

     QQ[x ..x ]
         0   1
o7 = ----------
        4    3
       x  - x
        0    1

o7 : QuotientRing

See also

Ways to use semigroupRing:

For the programmer

The object semigroupRing is a method function with options.