solveFrobeniusIdeal -- solving Frobenius ideals

Synopsis

Usage:

solveFrobeniusIdeal I

solveFrobeniusIdeal(I, W)
Inputs:
- I, an ideal, a Frobenius ideal which is m-primary
Outputs:
- a list, containing monomials times logarithms of the variables

Description

See [SST, Algorithm 2.3.14].

Here is [SST, Example 2.3.16]:

i1 : R = QQ[t_1..t_5];

i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);

o2 : Ideal of R

i3 : solveFrobeniusIdeal I
 -- 9.34e-06 seconds elapsed
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

                                                                             
o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
                0        1        2       3        0       1       2       4 
     ------------------------------------------------------------------------
     1             1    2   1             1             1             1    2
     -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
     4    1    0   8    1   4    2    1   4    3    0   4    3    2   8    3
     ------------------------------------------------------------------------
       1             1             1             3                 2
     - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
       2    4    0   4    4    1   2    4    2   4    4    3       4

o3 : List

i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);

i5 : solveFrobeniusIdeal(I, W)
 -- 7.93e-06 seconds elapsed
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

                                                                             
o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
                0        1        2       3        0       1       2       4 
     ------------------------------------------------------------------------
     1             1    2   1             1             1             1    2
     -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
     4    1    0   8    1   4    2    1   4    3    0   4    3    2   8    3
     ------------------------------------------------------------------------
       1             1             1             3                 2
     - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
       2    4    0   4    4    1   2    4    2   4    4    3       4

o5 : List

Ways to use solveFrobeniusIdeal :

solveFrobeniusIdeal(Ideal)
solveFrobeniusIdeal(Ideal,Ring)

For the programmer

The object solveFrobeniusIdeal is a method function.