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TestIdeals :: frobeniusRoot

frobeniusRoot -- computes I^[1/p^e] in a polynomial ring over a perfect field

Synopsis

Description

In a polynomial ring k[x1, ..., xn] with cofficients in a field of positive characteristic p, the Frobenius root I[1/pe] is the smallest ideal J such that I⊆J[pe] (= frobeniusPower(J,e) ). This function computes it. Alternately it can be viewed as the image under the Cartier operator of the ideal I.

There are many ways to call frobeniusRoot. The simplest way is to call frobeniusRoot(e,I). For instance,

i1 : R = ZZ/5[x,y,z];
i2 : I = ideal(x^50*z^95, y^100+z^27);

o2 : Ideal of R
i3 : frobeniusRoot(2, I)

                4
o3 = ideal (z, y )

o3 : Ideal of R

This computes I[1/pe], i.e. the pe-th root of I. Often, one wants to compute the frobeniusRoot of some product of ideals. This is best accomplished by calling the following version of frobeniusRoot:

i4 : R =  ZZ/5[x,y,z];
i5 : I1 = ideal(x^10, y^10, z^10);

o5 : Ideal of R
i6 : I2 = ideal(x^20*y^100, x + z^100);

o6 : Ideal of R
i7 : I3 = ideal(x^50*y^50*z^50);

o7 : Ideal of R
i8 : frobeniusRoot(1, {4,5,6}, {I1, I2, I3})

             64 80 68   64 82 66   66 80 66   64 84 64   66 82 64   68 80 64 
o8 = ideal (x  y  z  , x  y  z  , x  y  z  , x  y  z  , x  y  z  , x  y  z  ,
     ------------------------------------------------------------------------
      64 86 62   66 84 62   68 82 62   70 80 62   64 88 60   66 86 60 
     x  y  z  , x  y  z  , x  y  z  , x  y  z  , x  y  z  , x  y  z  ,
     ------------------------------------------------------------------------
      68 84 60   70 82 60   72 80 60   60 60 168    61 60 68   60 62 166  
     x  y  z  , x  y  z  , x  y  z  , x  y  z    + x  y  z  , x  y  z    +
     ------------------------------------------------------------------------
      61 62 66   62 60 166    63 60 66   60 64 164    61 64 64   62 62 164  
     x  y  z  , x  y  z    + x  y  z  , x  y  z    + x  y  z  , x  y  z    +
     ------------------------------------------------------------------------
      63 62 64   64 60 164    65 60 64   60 66 162    61 66 62   62 64 162  
     x  y  z  , x  y  z    + x  y  z  , x  y  z    + x  y  z  , x  y  z    +
     ------------------------------------------------------------------------
      63 64 62   64 62 162    65 62 62   66 60 162    67 60 62   60 68 160  
     x  y  z  , x  y  z    + x  y  z  , x  y  z    + x  y  z  , x  y  z    +
     ------------------------------------------------------------------------
      61 68 60   62 66 160    63 66 60   64 64 160    65 64 60   66 62 160  
     x  y  z  , x  y  z    + x  y  z  , x  y  z    + x  y  z  , x  y  z    +
     ------------------------------------------------------------------------
      67 62 60   68 60 160    69 60 60
     x  y  z  , x  y  z    + x  y  z  )

o8 : Ideal of R

The above example computes the ideal (I14 I25 I36)[1/p]. For legacy reasons, you can specify the last ideal in your list using frobeniusRoot(e,exponentList,idealList,I). This last ideal is just raised to the first power.

You can also call frobeniusRoot(e,a,f). This computes the eth root of the principal ideal (fa). Calling frobeniusRoot(e,m,I) computes the eth root of the ideal Im, and calling frobeniusRoot(e,a,f,I) computes the eth root of the product fa I. Finally, you can also compute the pe-th root of a matrix A by calling frobeniusRoot(e,A).

See also

Ways to use frobeniusRoot :