2022-11-18 15:00:34 +01:00
|
|
|
def naive_hensel(fct, F, start = 1, prec=10):
|
|
|
|
|
'''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.'''
|
|
|
|
|
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
|
|
|
|
RtQ = FractionField(Rt)
|
|
|
|
|
RptW.<W> = PolynomialRing(RtQ)
|
|
|
|
|
RptWQ = FractionField(RptW)
|
|
|
|
|
fct = RptWQ(fct)
|
|
|
|
|
fct = RptW(numerator(fct))
|
|
|
|
|
#return(fct)
|
|
|
|
|
#while fct not in RptW:
|
|
|
|
|
# print(fct)
|
|
|
|
|
# fct *= W
|
2024-06-10 19:55:49 +02:00
|
|
|
alpha = (fct.derivative())(W = RtQ(start))
|
2022-11-18 15:00:34 +01:00
|
|
|
w0 = Rt(start)
|
|
|
|
|
i = 1
|
|
|
|
|
while(i < prec):
|
|
|
|
|
w0 = w0 - fct(W = w0)/alpha + O(t^(prec))
|
|
|
|
|
i += 1
|
2024-01-03 10:06:35 +01:00
|
|
|
return w0
|
|
|
|
|
|
|
|
|
|
def nth_root2(fct, n, prec=10):
|
|
|
|
|
'''Given power series in F((t)), find its n-th root up to precision prec.'''
|
|
|
|
|
F= parent(fct).base_ring()
|
|
|
|
|
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
|
|
|
|
RW.<W> = PolynomialRing(Rt)
|
|
|
|
|
v = fct.valuation()
|
|
|
|
|
fct1 = Rt(fct*t^(-v))
|
|
|
|
|
a0 = fct1[0]
|
|
|
|
|
if v%n != 0:
|
|
|
|
|
raise ValueError('The valuation of the power series is not divisible by n.')
|
|
|
|
|
return t^(v//n)*naive_hensel(W^n - fct1, F, start = a0.nth_root(n), prec=prec)
|
