We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain $R$ of positive characteristic (for $N\ge 1$) or for any Dedekind domain $R$ of positive characteristic (but only for $N\ge 2$), we give a closed formula for a set $𝒞YCL(R,N)$ of all possible cycle-lengths for polynomial mappings in $R^N$. Then we give a new property of sets $𝒞YCL(R,1)$, which refutes a kind of conjecture posed by W. Narkiewicz.

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...