26 lines
989 B
Python
26 lines
989 B
Python
def dual_elt(AS, zmag):
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'''Find the trace dual of a given elt zmag in the function field of an Artin-Schreier cover AS.'''
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p = AS.characteristic
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n = AS.height
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G = AS.group
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variable_names = 'x, y'
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for i in range(n):
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variable_names += ', z' + str(i)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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M = matrix(RxyzQ, p^n, p^n)
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for i in range(p^n):
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for j in range(p^n):
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elt = (zmag.group_action(G[i])*zmag.group_action(G[j])).trace().function
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elt = Rxyz(elt.numerator())/Rxyz(elt.denominator())
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M[i, j] = RxyzQ(elt)
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main_det = M.determinant()
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zvee = as_function(AS, 0)
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for i in range(p^n):
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Mprim = matrix(RxyzQ, M)
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Mprim[:, i] = vector([(j == 0) for j in range(p^n)])
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fi = Mprim.determinant()/main_det
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zvee += as_function(AS, fi*(zmag.group_action(G[i]).function))
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return zvee |