42 lines
1.9 KiB
Python
42 lines
1.9 KiB
Python
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def artin_schreier_transform(power_series, prec = 10):
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"""Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
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-jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve
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z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
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correction = 0
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F = power_series.parent().base()
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p = F.characteristic()
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RtQ = FractionField(Rt)
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power_series = RtQ(power_series)
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if power_series.valuation() == +Infinity:
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raise ValueError("Precision is too low.")
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if power_series.valuation() >= 0:
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# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
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aux = t^p - t
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z = new_reverse(aux, prec = prec)
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z = z(t = power_series)
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return(0, 0, t, z)
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while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
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M = -power_series.valuation()/p
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coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
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correction += coeff.nth_root(p)*t^(-M)
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power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
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jump = max(-(power_series.valuation()), 0)
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try:
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if jump != 0:
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T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
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except:
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print("no ", str(jump), "-th root; divide by", power_series.list()[0])
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return (jump, power_series.list()[0])
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if jump != 0:
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T_rev = new_reverse(T, prec = prec)
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t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
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z = 1/t^(jump) + Rt(correction)(t = t_old)
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return(jump, correction, t_old, z)
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if jump == 0:
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aux = t^p - t
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z = new_reverse(aux, prec = prec)
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z = z(t = power_series)
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z = z + correction
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return(0, correction, t, z)
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