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GroebnerStrata :: GroebnerStrata

GroebnerStrata -- a package for creating families of ideals with the same initial ideal

Description

The GroebnerStrata package is designed to compute, given a monomial ideal in a polynomial ring, (and a term order, coming from the polynomial ring), a family of homogeneous ideals all with the given monomial ideal as its lead term (initial) ideal.

The GroebnerStrata package is designed for computing homogeneous strata and parameter families for monomial ideals in polynomial rings. In certain instances homogeneous strata can be used to compute local coordinates on Hilbert Schemes.

Here is an example of the basic use of the package. We compute the Groebner family of the ideal $(a^2, ab, b^2, ac)$. This is an ideal in a polynomial ring with variables the same as $S$, and whose coefficient ring is a polynomial ring containing all of the parameters. The groebnerStratum(Ideal) function returns the ideal in the parameters of the locus of parameters, for which the given family is a Groebner basis.

i1 : kk = ZZ/101;
i2 : S = kk[a..d];
i3 : M = ideal"a2,ab,b2,ac";

o3 : Ideal of S
i4 : F = groebnerFamily M;

o4 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]
                  6   5   12   2   4   11   18   24   1   3   8   10   17   23   7   9   14   16   20   22   13   15   19   21
i5 : netList F_*

     +-------------------------------------------------------+
     | 2                      2                      2       |
o5 = |a  + t b*c + t a*d + t c  + t b*d + t c*d + t d        |
     |      1       3       2      4       5       6         |
     +-------------------------------------------------------+
     |                         2                         2   |
     |a*b + t b*c + t a*d + t c  + t  b*d + t  c*d + t  d    |
     |       7       9       8      10       11       12     |
     +-------------------------------------------------------+
     | 2                         2                         2 |
     |b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d  |
     |      13       15       14      16       17       18   |
     +-------------------------------------------------------+
     |                            2                         2|
     |a*c + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d |
     |       19       21       20      22       23       24  |
     +-------------------------------------------------------+
i6 : J = trim groebnerStratum F

                                                                             
o6 = ideal (t  - t   + t  t  , t  t   - t  t  t  , t   - t   + t  t   - t t  
             7    20    13 19   22 15    15 19 21   10    23    22 13    9 19
     ------------------------------------------------------------------------
                    2                                                   
     + t  t   - t  t   + t  t   - t  t  t  , t  + t  t  , t  + 2t  t   -
        16 19    15 19    20 21    13 19 21   8    14 19   1     20 19  
     ------------------------------------------------------------------------
         2                                                            
     t  t  , t t   + 50t  t   - 50t  t  t   - t t  t   - 50t  t  t   +
      13 19   9 22      16 22      22 13 21    9 19 21      16 19 21  
     ------------------------------------------------------------------------
              2                                                        
     50t  t  t  , t   + 50t  t   - t  t   - 50t  t  t   - 50t  t  t   +
        13 19 21   24      16 22    23 21      22 13 21      16 19 21  
     ------------------------------------------------------------------------
         2             2          2                                      
     t  t   + 50t  t  t  , t   + t  - t t   - t t   + t  t   + t t  t   -
      20 21      13 19 21   18    9    9 16    3 15    23 15    9 15 19  
     ------------------------------------------------------------------------
                  2  2                                                   
     t  t  t   + t  t   - t  t   + t t  t   - 2t  t  t   + t  t  t  t   +
      16 15 19    15 19    17 21    9 13 21     20 15 21    13 15 19 21  
     ------------------------------------------------------------------------
         2                                                              
     t  t  , t   - t t   + t  t   + t  t   - t  t  t   - t  t  t  , t  +
      14 21   11    9 20    14 22    17 19    20 15 19    14 19 21   4  
     ------------------------------------------------------------------------
                                                  2        3                
     2t  t   - t t   + 2t  t   - 2t  t  t   - t  t   + t  t   - 4t  t  t   +
       20 22    3 19     23 19     22 13 19    16 19    15 19     20 19 21  
     ------------------------------------------------------------------------
          2            2        2                                      
     2t  t  t  , t  + t   - t  t  , t  t   + 50t  t  t   - 2t  t  t   -
       13 19 21   2    20    14 19   17 22      16 22 13     14 22 21  
     ------------------------------------------------------------------------
           2                                            2       2     2  
     50t  t  t   - t  t  t   - 50t  t  t  t   + 2t  t  t   + 50t  t  t  ,
        22 13 21    17 19 21      16 13 19 21     14 19 21      13 19 21 
     ------------------------------------------------------------------------
                        2                                          
     t t   - 2t  t   + t  t   + t  t  t   + 4t  t  t   - t t  t   +
      3 22     23 22    22 13    16 22 19     20 22 21    3 19 21  
     ------------------------------------------------------------------------
                                      2               2         2  2        
     2t  t  t   - 3t  t  t  t   - t  t  t   - 4t  t  t   + 2t  t  t  , t   -
       23 19 21     22 13 19 21    16 19 21     20 19 21     13 19 21   12  
     ------------------------------------------------------------------------
                            2                                        
     t  t  - 50t  t  t   - t t   + t t  t   + t t  t   - 2t  t  t   -
      23 9      16 22 13    9 19    9 16 19    3 15 19     23 15 19  
     ------------------------------------------------------------------------
           2           2     2  3                                 2      
     t t  t   + t  t  t   - t  t   + t t  t   + t  t  t   + 50t  t  t   +
      9 15 19    16 15 19    15 19    9 20 21    14 22 21      22 13 21  
     ------------------------------------------------------------------------
                                                                       2    
     t  t  t   - t t  t  t   + 50t  t  t  t   + 3t  t  t  t   - t  t  t  t  
      17 19 21    9 13 19 21      16 13 19 21     20 15 19 21    13 15 19 21
     ------------------------------------------------------------------------
               2       2     2                                           2   
     - 2t  t  t   - 50t  t  t  , t  - t t   + 2t  t   - 2t  t  t   - t  t   +
         14 19 21      13 19 21   5    3 20     23 20     14 22 19    17 19  
     ------------------------------------------------------------------------
            2      2            2                    2                    2 
     t  t  t   - 2t  t   + 2t  t  t  , t  - t t   + t   + t  t  t   - t  t  
      20 15 19     20 21     14 19 21   6    3 23    23    16 20 22    14 22
     ------------------------------------------------------------------------
                       2 2          2          2            2          3   
     - t  t  t  t   + t t   - t t  t   - t t  t   + 2t  t  t   + t t  t   -
        16 22 13 19    9 19    9 16 19    3 15 19     23 15 19    9 15 19  
     ------------------------------------------------------------------------
            3     2  4                                                       
     t  t  t   + t  t   + t t  t   - 2t  t  t   - t  t  t  t   - t  t  t  t  
      16 15 19    15 19    3 20 21     23 20 21    20 22 13 21    16 20 19 21
     ------------------------------------------------------------------------
           2              2             2              2               2    
     + t  t  t  t   - t  t  t   + t t  t  t   + t  t  t  t   - 3t  t  t  t  
        22 13 19 21    17 19 21    9 13 19 21    16 13 19 21     20 15 19 21
     ------------------------------------------------------------------------
              3        2  2              2         2  2     2  2  2
     + t  t  t  t   + t  t   + t  t  t  t   + 2t  t  t   - t  t  t  )
        13 15 19 21    20 21    20 13 19 21     14 19 21    13 19 21

o6 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ]
                  6   5   12   2   4   11   18   24   1   3   8   10   17   23   7   9   14   16   20   22   13   15   19   21

The ideal of the parameter space of all homogeneous ideals with this lead term ideal is an ideal in 24 variables. Often, these parameter ideals are in too many variables to easily analyze them. But in this case we can determine the irreducible components of the ideal $J$. There are two components, of dimensions 8 and 11. Note that they are both rational varieties.

i7 : compsJ = decompose J;
i8 : compsJ = compsJ/trim;
i9 : compsJ/dim

o9 = {11, 8}

o9 : List
i10 : netList compsJ_0_*

      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o10 = |t   - t  t                                                                                                                                                                                       |
      | 22    19 21                                                                                                                                                                                     |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  - t   + t  t                                                                                                                                                                                  |
      | 7    20    13 19                                                                                                                                                                                |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                 2                                                                                                                                                               |
      |t   - t   - t t   + t  t   - t  t   + t  t                                                                                                                                                       |
      | 10    23    9 19    16 19    15 19    20 21                                                                                                                                                     |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  + t  t                                                                                                                                                                                        |
      | 8    14 19                                                                                                                                                                                      |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                   2                                                                                                                                                                             |
      |t  + 2t  t   - t  t                                                                                                                                                                              |
      | 1     20 19    13 19                                                                                                                                                                            |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                   2                                                                                                                                                                             |
      |t   - t  t   + t  t                                                                                                                                                                              |
      | 24    23 21    20 21                                                                                                                                                                            |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |       2                                                    2  2                                                        2                                                                        |
      |t   + t  - t t   - t t   + t  t   + t t  t   - t  t  t   + t  t   - t  t   + t t  t   - 2t  t  t   + t  t  t  t   + t  t                                                                         |
      | 18    9    9 16    3 15    23 15    9 15 19    16 15 19    15 19    17 21    9 13 21     20 15 21    13 15 19 21    14 21                                                                       |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t   - t t   + t  t   - t  t  t                                                                                                                                                                   |
      | 11    9 20    17 19    20 15 19                                                                                                                                                                 |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                           2        3                                                                                                                                                            |
      |t  - t t   + 2t  t   - t  t   + t  t   - 2t  t  t                                                                                                                                                |
      | 4    3 19     23 19    16 19    15 19     20 19 21                                                                                                                                              |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2        2                                                                                                                                                                                 |
      |t  + t   - t  t                                                                                                                                                                                  |
      | 2    20    14 19                                                                                                                                                                                |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |               2                                               2           2     2  3                                                                2              2                            |
      |t   - t  t  - t t   + t t  t   + t t  t   - 2t  t  t   - t t  t   + t  t  t   - t  t   + t t  t   + t  t  t   - t t  t  t   + 3t  t  t  t   - t  t  t  t   - t  t  t                             |
      | 12    23 9    9 19    9 16 19    3 15 19     23 15 19    9 15 19    16 15 19    15 19    9 20 21    17 19 21    9 13 19 21     20 15 19 21    13 15 19 21    14 19 21                           |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                           2           2      2                                                                                                                                                  |
      |t  - t t   + 2t  t   - t  t   + t  t  t   - 2t  t                                                                                                                                                |
      | 5    3 20     23 20    17 19    20 15 19     20 21                                                                                                                                              |
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |              2     2 2          2          2            2          3           3     2  4                                2             2               2              3        2  2        2  2 |
      |t  - t t   + t   + t t   - t t  t   - t t  t   + 2t  t  t   + t t  t   - t  t  t   + t  t   + t t  t   - 2t  t  t   - t  t  t   + t t  t  t   - 3t  t  t  t   + t  t  t  t   + t  t   + t  t  t  |
      | 6    3 23    23    9 19    9 16 19    3 15 19     23 15 19    9 15 19    16 15 19    15 19    3 20 21     23 20 21    17 19 21    9 13 19 21     20 15 19 21    13 15 19 21    20 21    14 19 21|
      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i11 : netList compsJ_1_*

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o11 = |t                                                                                                                                                                                                                 |
      | 15                                                                                                                                                                                                               |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  + 50t   - 50t  t                                                                                                                                                                                               |
      | 9      16      13 21                                                                                                                                                                                             |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  - t   + t  t                                                                                                                                                                                                   |
      | 7    20    13 19                                                                                                                                                                                                 |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                              2                                                                                                                                                                                   |
      |t   + 50t  t   - 2t  t   - 50t  t                                                                                                                                                                                 |
      | 17      16 13     14 21      13 21                                                                                                                                                                               |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t   - t   + t  t   - 50t  t   + t  t   + 50t  t  t                                                                                                                                                                |
      | 10    23    22 13      16 19    20 21      13 19 21                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  + t  t                                                                                                                                                                                                         |
      | 8    14 19                                                                                                                                                                                                       |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  - 2t   + t  t   + t  t   + 4t  t   - 2t  t  t                                                                                                                                                                  |
      | 3     23    22 13    16 19     20 21     13 19 21                                                                                                                                                                |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                   2                                                                                                                                                                                              |
      |t  + 2t  t   - t  t                                                                                                                                                                                               |
      | 1     20 19    13 19                                                                                                                                                                                             |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                                          2             2                                                                                                                                         |
      |t   + 50t  t   - t  t   - 50t  t  t   - 50t  t  t   + t  t   + 50t  t  t                                                                                                                                          |
      | 24      16 22    23 21      22 13 21      16 19 21    20 21      13 19 21                                                                                                                                        |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |         2        2       2  2                                                                                                                                                                                    |
      |t   + 25t   - t  t   - 25t  t                                                                                                                                                                                     |
      | 18      16    14 21      13 21                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                                                     2                                                                                                                                            |
      |t   + 50t  t   + t  t   - 50t  t  t   - 50t  t  t   + t  t  t   + 50t  t  t                                                                                                                                       |
      | 11      16 20    14 22      16 13 19      20 13 21    14 19 21      13 19 21                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |t  + 2t  t   - t  t  t                                                                                                                                                                                            |
      | 4     20 22    22 13 19                                                                                                                                                                                          |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2        2                                                                                                                                                                                                  |
      |t  + t   - t  t                                                                                                                                                                                                   |
      | 2    20    14 19                                                                                                                                                                                                 |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                  2                                                     2                                 2       2     2                                                                         |
      |t   + 50t  t   - 50t  t  t   - 25t  t   - 50t  t  t   + t  t  t   - 50t  t  t   + 50t  t  t   + 50t  t  t  t   + 50t  t  t   - 25t  t  t                                                                          |
      | 12      23 16      16 22 13      16 19      16 20 21    14 22 21      23 13 21      22 13 21      16 13 19 21      20 13 21      13 19 21                                                                        |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                                                   2      2                          2  2                                                                                                                         |
      |t  + t  t  t   + t  t  t   - 2t  t  t   + 50t  t  t   + 2t  t   - 2t  t  t  t   - 50t  t  t                                                                                                                       |
      | 5    20 22 13    16 20 19     14 22 19      16 13 19     20 21     20 13 19 21      13 19 21                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                    2                                              2  2                                                                     2                 2         2  2               2       2  2  2 |
      |t  - t   + t  t  t   - t  t   + t  t  t   + t  t  t   - t  t  t  t   + 25t  t   + 4t  t  t   - 2t  t  t  t   - 2t  t  t  t   - 2t  t  t  t   + t  t  t  t   + t  t  t  t   - 3t  t   + 3t  t  t  t   - 26t  t  t  |
      | 6    23    16 20 22    14 22    23 22 13    23 16 19    16 22 13 19      16 19     23 20 21     20 22 13 21     16 20 19 21     23 13 19 21    22 13 19 21    16 13 19 21     20 21     20 13 19 21      13 19 21|
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

This tells us that there are 2 components (at least over the given field). Their dimensions are 11, 8.

We can find random points on each component, since these components are rational.

i12 : pt1 = randomPointOnRationalVariety compsJ_0

o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13
      -----------------------------------------------------------------------
      -36 -30 -29 -10 |

               1        24
o12 : Matrix kk  <--- kk
i13 : pt2 = randomPointOnRationalVariety compsJ_1

o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39
      -----------------------------------------------------------------------
      -16 0 -47 21 |

               1        24
o13 : Matrix kk  <--- kk
i14 : F1 = sub(F, (vars S)|pt1)

              2             2                              2               
o14 = ideal (a  + 2b*c - 32c  - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c -
      -----------------------------------------------------------------------
         2                              2   2             2                  
      30c  - 29a*d - 23b*d + 30c*d + 41d , b  - 36b*c - 8c  - 30a*d - 22b*d +
      -----------------------------------------------------------------------
                 2                   2                              2
      19c*d + 48d , a*c - 29b*c + 24c  - 10a*d - 13b*d + 19c*d + 36d )

o14 : Ideal of S
i15 : F2 = sub(F, (vars S)|pt2)

              2              2                              2               
o15 = ideal (a  - 31b*c + 26c  - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c -
      -----------------------------------------------------------------------
         2                             2   2              2                  
      16c  - 17a*d - 7b*d + 31c*d + 11d , b  - 16b*c + 19c  + 34b*d - 41c*d +
      -----------------------------------------------------------------------
         2                   2                              2
      38d , a*c - 47b*c - 38c  + 21a*d + 39b*d - 24c*d + 24d )

o15 : Ideal of S
i16 : decompose F1

                                    2             2                      2
o16 = {ideal (a - 29b + 24c - 44d, b  - 36b*c - 8c  + 17b*d + 32c*d + 41d ),
      -----------------------------------------------------------------------
      ideal (c - 10d, b + 33d, a + 10d)}

o16 : List
i17 : decompose F2

o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c -
      -----------------------------------------------------------------------
      29d)}

o17 : List

Note, the general element of one component is a plane conic union a point. (The dimension of the locus of all such is: (choice of plane) + (choice of degree 2 in plane) + choice of point. This is 3 + 5 + 3 = 11.

The other component consists of two skew lines. This has dimension (choice of line) + (choice of line). This is 4 + 4 = 8. Also notice that the 2 skew lines do not have to be defined over the base field, as in this case.

Authors

Version

This documentation describes version 0.9 of GroebnerStrata.

Source code

The source code from which this documentation is derived is in the file GroebnerStrata.m2.

Exports

  • Functions and commands
  • Methods
    • "findWeightConstraints(Ideal,List)" -- see findWeightConstraints -- returns a matrix of weight constraints
    • "findWeightVector(Ideal,List)" -- see findWeightVector -- returns a weight vector
    • "groebnerFamily(Ideal)" -- see groebnerFamily -- computes families of ideals with a specified initial ideal
    • "groebnerFamily(Ideal,List)" -- see groebnerFamily -- computes families of ideals with a specified initial ideal
    • "groebnerStratum(Ideal)" -- see groebnerStratum -- compute the ideal where a given is a Groebner basis
    • "nonminimalMaps(Ideal)" -- see nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal
    • randomPointOnRationalVariety(Ideal) -- find a random point on a variety that can be detected to be rational
    • randomPointsOnRationalVariety(Ideal,ZZ) -- find random points on a variety that can be detected to be rational
    • "smallerMonomials(Ideal)" -- see smallerMonomials -- returns the standard monomials smaller but of the same degree as given monomial(s)
    • "smallerMonomials(Ideal,RingElement)" -- see smallerMonomials -- returns the standard monomials smaller but of the same degree as given monomial(s)
    • "standardMonomials(Ideal)" -- see standardMonomials -- computes standard monomials
    • "standardMonomials(List,Ideal)" -- see standardMonomials -- computes standard monomials
    • "standardMonomials(ZZ,Ideal)" -- see standardMonomials -- computes standard monomials
    • "tailMonomials(Ideal)" -- see tailMonomials -- find tail monomials
    • "tailMonomials(Ideal,RingElement)" -- see tailMonomials -- find tail monomials
  • Symbols
    • AllStandard -- boolean option for determining the use of all standard or smaller standard monomials
    • Minimalize -- boolean option for determining whether excess parameters will be elimnated

For the programmer

The object GroebnerStrata is a package.