TITLE: What is the difference between Hensel lifting and the Newton-Raphson method?
QUESTION [9 upvotes]: So in the Newton-Raphson method to iteratively approximate a root of a real polynomial, we start with a crude approximation $x_0 \in \mathbb{R}$ for $f(x)=0$ where $f(x) \in \mathbb{R}[x]$. For the next iterate $x_1$, we put $x_1 = x_0 + \epsilon$, and we want to determine $\epsilon$ to get a better approximation. For this we use a Taylor series and take a linear approximation, and equate $f(x_1)$ to 0 to get a value of $\epsilon$.
$$ f(x_1) = f(x_0 + \epsilon) = f(x_0) + \epsilon f'(x_0) + O(\epsilon^2) \approx f(x_0) + \epsilon f'(x_0)$$
$$ 0 = f(x_0) + \epsilon f'(x_0)$$
$$ \epsilon = - \frac{f(x_0)}{f'(x_0)}$$
$$ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$$
Of course, a necessary condition here is that $f'(x_0) \neq 0$.
Now in the case of Hensel lifting of a root modulo $p$ of $f(x) \in \mathbb{Z}[x]$ to a root modulo $p^2$, we do something very similar. If $x_0 \in \mathbb{Z}$ is such that $f(x_0) \equiv 0 \pmod p$ and $f'(x_0) \neq 0 \pmod p$, then again we take $x_1 = x_0 + p\epsilon$, ignore everything but first order terms by going modulo $p^2$ and find $\epsilon$ by equating $f(x_1)$ to zero.
$$ f(x_1) = f(x_0 + p\epsilon) = f(x_0) + p\epsilon f'(x_0) + O(p^2\epsilon^2) $$
$$ 0 = f(x_0) + p\epsilon f'(x_0) \pmod {p^2} $$
$$ \epsilon = - \frac{f(x_0)}{pf'(x_0)}$$
$$ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$$
So just as before, we end up ignoring terms beyond the linear term, and the $x_1$ that we get is pretty much the same thing, except that if we do get a fraction, we think of it (the division or inverse) as operating within the ring $\mathbb{Z}/p^2\mathbb{Z}$ and express it as an integer.
So except for the starting value (in Newton-Raphson we start at some (any) crude approximation, while in Hensel lifting we start with a root modulo prime $p$), are the two methods essentially the same? Can't we obtain the lifts modulo higher powers of $p$ simply by looking at the iterates in the Newton-Raphson method, if only we start with an agreeable candidate?

REPLY [4 votes]: They are, from a distance, the same. The main difference is that the iterates of Newton's method always remain approximations, while Hensel lifting is exact in the values it computes. As p-adic numbers, the iterates of Hensel are also approximations, but their lower order terms become stationary after finitely many iterations.
So you can for instance use Hensel lifting to compute a power series expansion of some solution of an algebraic problem, and if this power series is actually a rational function, this can be exactly recovered from the Hensel-produced power series.