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

## Synopsis

• 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, , with entries in a ring of characteristic $p > 0$
• Optional inputs:
• FrobeniusRootStrategy => , 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)
• , 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).