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TestIdeals :: frobenius

frobenius -- computes Frobenius powers of ideals and matrices

Synopsis

Description

Given an ideal I in a ring of characteristic p > 0 and a nonnegative integer e, frobenius(e,I) or frobeniuse(I) returns the pe-th Frobenius power I[pe], that is, the ideal generated by all powers fpe, with f ∈I (see frobeniusPower).

i1 : R = ZZ/3[x,y];
i2 : I = ideal(x^2,x*y,y^2);

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

             18   9 9   18
o3 = ideal (x  , x y , y  )

o3 : Ideal of R
i4 : frobenius^2(I)

             18   9 9   18
o4 = ideal (x  , x y , y  )

o4 : Ideal of R
i5 : frobeniusPower(3^2,I)

             18   9 9   18
o5 = ideal (x  , x y , y  )

o5 : Ideal of R

If e is negative, then frobenius(e,I) or frobeniuse(I) computes a Frobenius root, as defined by Blickle, Mustata, and Smith (see frobeniusRoot).

i6 : R = ZZ/5[x,y,z,w];
i7 : I = ideal(x^27*y^10+3*z^28-x^2*y^15*z^35,x^17*w^30+2*x^10*y^10*z^35,x*z^50);

o7 : Ideal of R
i8 : frobenius(-1,I)

             5   5 2   3 6
o8 = ideal (z , x y , x w )

o8 : Ideal of R
i9 : frobenius(-2,I)

o9 = ideal (w, z, x)

o9 : Ideal of R
i10 : frobeniusRoot(2,I)

o10 = ideal (w, z, x)

o10 : Ideal of R

If M is a matrix with entries in a ring of positive characteristic p > 0 and e is a nonnegative integer, then frobenius(e,M) or frobeniuse(M) outputs a matrix whose entries are the pe-th powers of the entries of M.

i11 : M = ZZ/3[x,y];
i12 : M = matrix {{x,y},{x+y,x^2+y^2}};

              ZZ       2       ZZ       2
o12 : Matrix (--[x, y])  <--- (--[x, y])
               3                3
i13 : frobenius(2,M)

o13 = | x9    y9      |
      | x9+y9 x18+y18 |

              ZZ       2       ZZ       2
o13 : Matrix (--[x, y])  <--- (--[x, y])
               3                3

frobenius(I) and frobenius(M) are abbreviations for frobenius(1,I) and frobenius(1,M).

See also