31 lines
1.1 KiB
Python
31 lines
1.1 KiB
Python
def naive_hensel(fct, F, start = 1, prec=10):
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'''given field F and polynomial fct over F((t)), find root of this polynomial in F((t)), using Hensel method with first value equal to start.'''
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RtQ = FractionField(Rt)
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RptW.<W> = PolynomialRing(RtQ)
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RptWQ = FractionField(RptW)
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fct = RptWQ(fct)
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fct = RptW(numerator(fct))
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#return(fct)
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#while fct not in RptW:
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# print(fct)
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# fct *= W
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alpha = (fct.derivative())(W = start)
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w0 = Rt(start)
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i = 1
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while(i < prec):
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w0 = w0 - fct(W = w0)/alpha + O(t^(prec))
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i += 1
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return w0
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def nth_root2(fct, n, prec=10):
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'''Given power series in F((t)), find its n-th root up to precision prec.'''
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F= parent(fct).base_ring()
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RW.<W> = PolynomialRing(Rt)
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v = fct.valuation()
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fct1 = Rt(fct*t^(-v))
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a0 = fct1[0]
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if v%n != 0:
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raise ValueError('The valuation of the power series is not divisible by n.')
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return t^(v//n)*naive_hensel(W^n - fct1, F, start = a0.nth_root(n), prec=prec) |