Strong form of Hensel lemma

Let $K$ be complete for a discrete valuation $v$. If $f\in {\mathcal{O}}_K[X]$ andthen Newton iteration over a valued field converges to a root $\alpha \in {\mathcal{O}}_K$ with $v(\alpha -a)>v(f^{\prime }(a))$.

Let $R$ be a complete discrete valuation ring, $f\in R[X]$, and $d=|f^{\prime }(a)|$ satisfy $0<d<1$ and $|f(a)|=d^2$. Suppose the second Taylor coefficient of $f$ at every point of $R$ has norm below one. One Newton iteration over a valued field step has correction $h=-f(a)/f^{\prime }(a)$ of norm $d$. Its residual has quadratic term of norm strictly below $d^2$, and all higher terms have norm at most $d^3<d^2$. The derivative changes by a term of norm below $d$, so its norm remains $d$. The strong form of Hensel lemma therefore applies from the new point, producing a root within distance $d$ of the original point. The second Taylor coefficient is $f^{\prime \prime }/2$ when that quotient is meaningful, or the division-free binomial-coefficient Taylor expression in arbitrary characteristic. A Newton sequence that reaches a root is thereafter constant; residuals decrease strictly only before that event.

Starting from $a_0$, Newton iteration sets $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Under the strong form of Hensel lemma inequality, the correction valuations grow at least geometrically, so $(a_n)$ is a Cauchy sequence converging to a root.