cssLeadTerm -- lead term of the canonical series solutions of I

Synopsis

Usage:

cssLeadTerm(I, w)
Inputs:
- I, an ideal, (regular) holonomic ideal in the Weyl algebra
- w, a list, (generic) weights for $I$, of length half the number of variables in the Weyl algebra
Outputs:
- a list, containing monomials times logarithms in the variables

Description

This routine returns the lead terms of the canonical series solutions of $I$ with respect to the weight vector $w$. See [SST, Algorithm 2.3.14 and Lemma 2.5.10].

Here is [SST, Example 2.3.16]:

i1 : needsPackage "FourTiTwo"

o1 = FourTiTwo

o1 : Package

i2 : A = matrix{{1,1,1,1,1,1},{-2,0,0,0,0,1},{0,1,0,1,0,0},{1,1,2,0,0,1}}

o2 = | 1  1 1 1 1 1 |
     | -2 0 0 0 0 1 |
     | 0  1 0 1 0 0 |
     | 1  1 2 0 0 1 |

              4       6
o2 : Matrix ZZ  <-- ZZ

i3 : beta = {2,1,0,2}

o3 = {2, 1, 0, 2}

o3 : List

i4 : Hbeta = gkz(A,beta)

                                                                            
o4 = ideal (x D  + x D  + x D  + x D  + x D  + x D  - 2, - 2x D  + x D  - 1,
             1 1    2 2    3 3    4 4    5 5    6 6          1 1    6 6     
     ------------------------------------------------------------------------
                                                       2    2           2  
     x D  + x D , x D  + x D  + 2x D  + x D  - 2, - D D  + D D , - D D D  +
      2 2    4 4   1 1    2 2     3 3    6 6         3 4    2 5     2 3 5  
     ------------------------------------------------------------------------
          2     2            2     3 3    2 4
     D D D , - D D D  + D D D , - D D  + D D )
      1 4 6     3 4 5    1 2 6     3 5    1 6

o4 : Ideal of QQ[x ..x , D ..D ]
                  1   6   1   6

i5 : w = {9,1,99999, 9999999, 3, 999}

o5 = {9, 1, 99999, 9999999, 3, 999}

o5 : List

i6 : netList cssLeadTerm(Hbeta, w)
 -- 6.44e-06 seconds elapsed
 -- 6.69e-06 seconds elapsed
 -- 6.7e-06 seconds elapsed
 -- 9.151e-06 seconds elapsed
 -- 6.591e-06 seconds elapsed
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

     +----------------------------------------------------+
     |   1 5   5 5                                        |
     | - - - - - -                                        |
     |   2 2   2 2                                        |
o6 = |x   x x   x                                         |
     | 1   2 4   5                                        |
     +----------------------------------------------------+
     | -1                                                 |
     |x  x x x                                            |
     | 2  3 4 6                                           |
     +----------------------------------------------------+
     | -1       1        3        3                       |
     |x  x x x (-logX  - -logX  - -logX  + logX )         |
     | 2  3 4 6 2    0   4    2   4    4       5          |
     +----------------------------------------------------+
     |   -1                                               |
     |x x  x x                                            |
     | 2 4  5 6                                           |
     +----------------------------------------------------+
     |   -1     1        3        3        3              |
     |x x  x x (-logX  - -logX  + -logX  - -logX  + logX )|
     | 2 4  5 6 2    0   2    1   2    3   2    4       5 |
     +----------------------------------------------------+
     | -1 2 2 -1                                          |
     |x  x x x                                            |
     | 1  3 5 6                                           |
     +----------------------------------------------------+
     | -1 2 2 -1 1        3        3                      |
     |x  x x x  (-logX  - -logX  - -logX  + logX )        |
     | 1  3 5 6  2    0   4    2   4    4       5         |
     +----------------------------------------------------+
     | 1 5                                                |
     | - -                                                |
     | 3 3                                                |
     |x x                                                 |
     | 1 6                                                |
     +----------------------------------------------------+
     | 1 5                                                |
     | - -                                                |
     | 3 3 1        3        3                            |
     |x x (-logX  - -logX  - -logX  + logX )              |
     | 1 6 2    0   4    2   4    4       5               |
     +----------------------------------------------------+
     |   1     1                                          |
     | - -     -                                          |
     |   3     3                                          |
     |x   x x x                                           |
     | 1   3 5 6                                          |
     +----------------------------------------------------+
     |   1     1                                          |
     | - -     -                                          |
     |   3     3 1        3        3                      |
     |x   x x x (-logX  - -logX  - -logX  + logX )        |
     | 1   3 5 6 2    0   4    2   4    4       5         |
     +----------------------------------------------------+

Ways to use cssLeadTerm :

cssLeadTerm(Ideal,List)

For the programmer

The object cssLeadTerm is a method function.