33 lines
1.3 KiB
Python
33 lines
1.3 KiB
Python
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def quaternion_reduction(AS, fct):
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'''Simplify rational function fct as a function in the function field of AS, so that z[i] appear in powers <p and only in numerator'''
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n = AS.height
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F = AS.base_ring
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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p = F.characteristic()
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ff = AS.functions
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ff = [RxyzQ(F.function) for F in ff]
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fct = RxyzQ(fct)
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fct1 = numerator(fct)
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fct2 = denominator(fct)
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denom = quaternion_function(AS, fct2)
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denom_norm = prod(quaternion_function(AS, fct2).group_action(g) for g in AS.group if list(g) != n*[0])
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fct1 = Rxyz(fct1*denom_norm.function)
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fct2 = Rxyz(fct2*denom_norm.function)
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if fct2 != 1:
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return quaternion_reduction(AS, fct1)/quaternion_reduction(AS, fct2)
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result = RxyzQ(0)
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change = 0
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for a in fct1.monomials():
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degrees_zi = [a.degree(z[i]) for i in range(n)]
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d_div = [a.degree(z[i])//p for i in range(n)]
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if d_div != n*[0]:
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change = 1
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d_rem = [a.degree(z[i])%p for i in range(n)]
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monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n))
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result += RxyzQ(monomial)
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if change == 0:
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return RxyzQ(result)
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else:
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return quaternion_reduction(AS, RxyzQ(result))
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