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).