Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all integer polynomials $q$ such that $q$ has all its roots in $R_S$.

Can one (nicely) characterize $\hat{S}$ in terms of $S$? Sorry to be vague, but a nice characterization for me should stay in the domain of integer arithmetic, so express itself in terms of polynomial coefficients without an appeal to the complex numbers. (Perhaps someone knows a paradigm allows me to ask this more objectively?)

Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all integer polynomials $q$ such that $q$ has all its roots in $R_S$.

Can one (nicely) characterize $\hat{S}$ in terms of $S$? Sorry to be vague, but a nice characterization for me should stay in the domain of integer arithmetic, so express itself in terms of polynomial coefficients without an appeal to the complex numbers. (Perhaps someone knows a paradigm that allows me to ask this more objectively?)