= Boundary case of Hensel's lemma with a small quadratic coefficient
{c}
{title2=$|f(a)|=|f'(a)|^2$}
Let $R$ be a complete <discrete valuation ring>, $f\in R[X]$, and $d=|f'(a)|$ 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 $h=-f(a)/f'(a)$ 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 $f''/2$ 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.
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