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For any field $K$ equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton’s method applied to a squarefree polynomial $f\in K\left[x\right]$ will succeed in finding some root of $f$ in the $v$-adic topology for infinitely many places $v$ of $K$. Furthermore, we show that if $K$ is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge $v$-adically...

Let $f$ be a polynomial of degree at least 2 with coefficients in a number field $K$, let $x_0$ be a sufficiently general element of $K$, and let $\alpha$ be a root of $f$. We give precise conditions under which Newton iteration, started at the point $x_0$, converges $v$-adically to the root $\alpha$ for infinitely many places $v$ of $K$. As a corollary we show that if $f$ is irreducible over $K$ of degree at least 3, then Newton iteration converges $v$-adically to any given root of $f$ for infinitely many places $v$. We also conjecture that...