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defeo/Frobenius.ipynb

Created February 21, 2016 21:57
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A demonstration of Lemma 1.1
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Frobenius.ipynb
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schost commented Feb 21, 2016 

Q1.<xx>=R.quo(P1)
R1.<z>=PolynomialRing(Q1)
Q2.<zz>=R1.quo(z^(l^2)-xx)
xx = Q2(R1(xx))

def to_Q2(aa):
    return add([aa[i]*xx^(i//25)*zz^(i%25) for i in range(100)])

def from_Q1(b):
    return b.lift()(x^25)

def from_Q2(bb):
    return add([from_Q1(bb[i])*x^i for i in range(25)])

A = to_Q2(a)
check = from_Q2(A^sigma)
check == expected

YY = xx^u*zz^r
YY == zz^sigma

monomials = [Q2(1)]
for i in range(25):
    monomials.append(monomials[i]*YY)
## here, we should be a bit careful; these guys can all be computed in O(25) ops in F1.

res = add([monomials[i] * Q2(R1(A[i])) for i in range(25)])
## same, this sum takes O(25) ops in F1.

from_Q2(res) == expected

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