Newton approximation fails for 100% of the primes

Let $f$ be a polynomial of degree $d\geq 2$ with coefficients in a number field $K$, and let $x_0\in K$. Define the Newton map

$$N(t)=t-\frac{f(t)}{f'(t)},$$

and, for each $n\geq 0$, set

$$x_{n+1}=N(x_n).$$

Assume that the Newton approximation sequence $(x_n)$ is not eventually periodic. Let $C(K,f,x_0)$ be the set of places $v$ of $K$ for which $(x_n)$ converges $v$-adically to a root of $f$. Newton approximation fails for 100% of the primes. The natural density of $C(K,f,x_0)$ is zero. This gives a precise form of the paper's heuristic that convergence places are sparse; the statement is presented as a conjecture and no resolution is supplied here.