42 lines
No EOL
1.4 KiB
Python
42 lines
No EOL
1.4 KiB
Python
# Finding roots of polynomials in p-adic integers using Hensel's lemma
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from padic import *
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from poly import *
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def roots(p, poly):
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'Yield all roots of polynomial in the given p-adic integers.'
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for root in xrange(p):
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try:
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yield PAdicPoly(p, poly, root)
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except ValueError:
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pass
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class PAdicPoly(PAdic):
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'Result of lifting a root of a polynomial in the integers mod p to the p-adic integers.'
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def __init__(self, p, poly, root):
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PAdic.__init__(self, p)
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self.root = root
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self.poly = poly
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self.deriv = derivative(poly)
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# argument checks for the algorithm to work
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if poly(root) % p:
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raise ValueError("%d is not a root of %s modulo %d" % (root, poly, p))
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if self.deriv(root) % p == 0:
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raise ValueError("Polynomial %s is not separable modulo %d" % (poly, p))
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# take care of trailing zeros
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digit = self.root
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self.val = str(digit)
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self.exp = self.p
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while digit == 0:
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self.order += 1
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digit = self._nextdigit()
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# self.prec += 1
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def _nextdigit(self):
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self.root = ModP(self.exp * self.p, self.root)
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self.root = self.root - self.poly(self.root) / self.deriv(self.root) # coercions automatically taken care of
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digit = self.root // self.exp
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self.exp *= self.p
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return digit |