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Hensel's Lemma

richardhp
Post Icon Posted: Submitted by richardhp on 29 December 2009 - 1:02pm.

Joined: 2007-10-01
Posts: 239

Ok so I read on wikipedia I think that Hensel's lemma is somehow a generalisation of the Newton-Raphson method. I've been thinking about this for a bit and I can't really see where they're coming from on this one.

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Sam
Post Icon Posted: 30 December 2009 - 2:14pm

Joined: 2007-10-03
Posts: 562

The Newton-Raphson method uses the formula $ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $ to give you a better approximation to a root of your polynomial; here we are working with real numbers and the usual absolute value metric. Clearly we need $ f'(x_n) $ to be nonzero, and even then the method might still fail...

Hensel's lemma does essentially the same thing for $ p $-adic numbers: say you have a solution $ x_n $ satisfying $ f(x_n) \equiv 0 \,\, (\text{mod}\ p^n) $. You can view this as an approximation to a root of $ f $ in the $ p $-adic numbers, using the $ p $-adic metric.
Then $ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $ is a better approximation to a $ p $-adic root: $ f(x_{n+1}) \equiv 0 \,\, (\text{mod}\ p^{n+1}) $. Then you can iterate this, and it will converge to a root in the ring of $ p $-adic integers. This method never fails, all you need is $ f'(x_0) $ to be invertible mod $ p $.

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