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GroebnerStrata :: groebnerFamily

groebnerFamily -- computes families of ideals with a specified initial ideal

Synopsis

Description

Given a monomial ideal $M$ in a polynomial ring $R$, this computes the parameter families of homogeneous ideals where $M$ could be their initial ideal. These families are obtained from either the standard monomials to the generators of $M$, or the standard monomials smaller than the generators of $M$ but of the same degree as these generators. In the former case we obtain a family of all ideals where $M$ could be their initial ideal. In the latter case, we obtain such a family with respect to a given term order.

i1 : R = ZZ/32003[a,b,c,d];
i2 : M = ideal (a^2, a*b, b^2)

             2        2
o2 = ideal (a , a*b, b )

o2 : Ideal of R
i3 : F = groebnerFamily M

             2                                      2              2       
o3 = ideal (a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d , a*b +
                  1       2       4       5       3      6       7         
     ------------------------------------------------------------------------
                                           2                2   2           
     t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d , b  + t  a*c +
      8       9       11       12       10      13       14          15     
     ------------------------------------------------------------------------
                                    2                2
     t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d )
      16       18       19       17      20       21

                ZZ
o3 : Ideal of -----[t , t ..t , t  , t  ..t  , t  , t  ..t  , t ..t , t ..t , t ..t , t  ..t  , t  ..t  , t  ..t  ][a..d]
              32003  3   6   7   10   13   14   17   20   21   1   2   4   5   8   9   11   12   15   16   18   19
i4 : netList F_*

     +---------------------------------------------------------------+
     | 2                                      2              2       |
o4 = |a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d        |
     |      1       2       4       5       3      6       7         |
     +---------------------------------------------------------------+
     |                                            2                2 |
     |a*b + t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d  |
     |       8       9       11       12       10      13       14   |
     +---------------------------------------------------------------+
     | 2                                           2                2|
     |b  + t  a*c + t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d |
     |      15       16       18       19       17      20       21  |
     +---------------------------------------------------------------+
i5 : U = ring F

o5 = U

o5 : PolynomialRing
i6 : T = coefficientRing U

o6 = T

o6 : PolynomialRing
i7 : gens T

o7 = {t , t , t , t  , t  , t  , t  , t  , t  , t , t , t , t , t , t , t  ,
       3   6   7   10   13   14   17   20   21   1   2   4   5   8   9   11 
     ------------------------------------------------------------------------
     t  , t  , t  , t  , t  }
      12   15   16   18   19

o7 : List
i8 : gens U

o8 = {a, b, c, d}

o8 : List

Here, $F$ is the family of homogeneous ideals having $M$ as their initial ideal, under the term order of the ring of $M$.

The optional argument AllStandard is boolean, taking the value $true$ to compute the family of all homogeneous ideals with a given initial ideal and the value $false$ to compute the family with respect to a given order. The default value for this argument is false.

If $L$ is not given, then it is computed using standardMonomials (if AllStandard is true), or smallerMonomials (if AllStandard is false).

i9 : L = standardMonomials M

                  2                  2               2                  2  
o9 = {{a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d, d },
     ------------------------------------------------------------------------
                 2                  2
     {a*c, b*c, c , a*d, b*d, c*d, d }}

o9 : List
i10 : F2 = groebnerFamily (M, L)

              2                                      2              2       
o10 = ideal (a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d , a*b +
                   1       2       4       5       3      6       7         
      -----------------------------------------------------------------------
                                            2                2   2           
      t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d , b  + t  a*c +
       8       9       11       12       10      13       14          15     
      -----------------------------------------------------------------------
                                     2                2
      t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d )
       16       18       19       17      20       21

                 ZZ
o10 : Ideal of -----[t , t ..t , t  , t  ..t  , t  , t  ..t  , t ..t , t ..t , t ..t , t  ..t  , t  ..t  , t  ..t  ][a..d]
               32003  3   6   7   10   13   14   17   20   21   1   2   4   5   8   9   11   12   15   16   18   19

Note that $F$ and $F_2$ are the same family, in this case.

See also

Ways to use groebnerFamily :

For the programmer

The object groebnerFamily is a method function with options.