  Let $E$ be an elliptic curve with positive rank over a number field $K$ and
let $p$ be an odd prime number. Let $K_{cyc}$ be the cyclotomic
$\mathbb{Z}_p$-extension of $K$ and $K_n$ denote its $n$-th layer. The
Mordell--Weil rank of $E$ is said to be constant in the cyclotomic tower of $K$
if for all $n$, the rank of $E(K_n)$ is equal to the rank of $E(K)$. We apply
techniques in Iwasawa theory to obtain explicit conditions for the rank of an
elliptic curve to be constant in the above sense. We then indicate the
potential applications to Hilbert's tenth problem for number rings.
