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
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i4 : frobenius^2(I)
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o4 = ideal (x , x y , y )
o4 : Ideal of R
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i5 : frobeniusPower(3^2,I)
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o5 = ideal (x , x y , y )
o5 : Ideal of R
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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)
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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])
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i13 : frobenius(2,M)
o13 = | x9 y9 |
| x9+y9 x18+y18 |
ZZ 2 ZZ 2
o13 : Matrix (--[x, y]) <--- (--[x, y])
3 3
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frobenius(I) and frobenius(M) are abbreviations for frobenius(1,I) and frobenius(1,M).