frobenius -- compute a Frobenius power of an ideal or a matrix

Usage:

frobenius(e, I)

frobenius^e(I)

frobenius(e, M)

frobenius^e(M)

frobenius(I)

frobenius(M)
Inputs:
- e, an integer, the power of Frobenius to take
- I, an ideal, in a ring of characteristic $p > 0$
- M, a matrix, with entries in a ring of characteristic $p > 0$
Optional inputs:
- FrobeniusRootStrategy => a symbol, default value Substitution, selects the strategy for internal frobeniusRoot calls
Outputs:
- an ideal, the p^e-th Frobenius power of I (with e = 1, if e is not specified)
- a matrix, the p^e-th Frobenius power of M (with e = 1, if e is not specified)

Description

Given an ideal $I$ in a ring of characteristic $p > 0$ and a nonnegative integer $e$, frobenius(e, I) or frobenius^e(I) returns the $p^e$-th Frobenius power $I^{[p^e]}$, that is, the ideal generated by all powers $f^{ p^e}$, with $f \in\ I$ (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 frobenius^e(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 characteristic $p > 0$ and $e$ is a nonnegative integer, then frobenius(e, M), or frobenius^e(M), outputs a matrix whose entries are the $p^e$-th powers of the entries of $M$.

i11 : R = ZZ/3[x,y];

i12 : M = matrix {{x, y},{x + y, x^2 + y^2}};

              2      2
o12 : Matrix R  <-- R

i13 : frobenius(2, M)

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

              2      2
o13 : Matrix R  <-- R

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

For the programmer

The object frobenius is an instance of the type FrobeniusOperator.

The source of this document is in TestIdeals/frobeniusPowersDoc.m2:63:0.

frobenius -- compute a Frobenius power of an ideal or a matrix

Description

See also

For the programmer