Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

We show that for any quadratic extension of number fields $K\mathrm{/}F$, there exists an abelian variety $A\mathrm{/}F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring ${\mathcal{O}}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over ${\mathcal{O}}_K$ has solutions in ${\mathcal{O}}_K$.