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# 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 .