Divisibility properties of polynomial expressions of random integers

We study divisibility properties of a set $\{f_1({\mathbf{U}}_n^{(s)}),\dots ,f_m({\mathbf{U}}_n^{(s)})\}$, where $f_1,\dots ,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and ${\mathbf{U}}_n^{(s)}$ is a point picked uniformly at random from the set $\{1,\dots ,n{\}}^s$, $s\in \mathbb{N}$. We show that the $\mathrm{G}\mathrm{C}\mathrm{D}$ and the suitably normalized $\mathrm{L}\mathrm{C}\mathrm{M}$ of this set converge in distribution to a.s.\ finite random variables under mild assumptions on $f_1,\dots ,f_m$. Our approach is based on the notion of integer adeles and a known fact that the uniform distribution on $\{1,\dots ,n\}$ converges to the Haar measure on the ring of integer adeles combined with the Lang-Weil bounds.