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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?)

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  • $\begingroup$ What has come up in various places is that one associates with a polynomial the probability measure on $\mathbb C$, which is equidistributed on the roots of the polynomial. Now, the space of probability measures has a natural topology and limit points are usually very interesting and special; there are isolated points etc. Equilibrium measures of compact subsets arise this way. $\endgroup$
    – Andreas Thom
    Commented Feb 21, 2012 at 7:54
  • $\begingroup$ Do you have an idea of the answer for the (probably simpler) case of polynomials over $\mathbb{C}$? $\endgroup$
    – Jérôme Poineau
    Commented Feb 21, 2012 at 19:33
  • $\begingroup$ @Jérôme A special case of your suggestion amounts to finding the degree 1 polynomials in "closure," which just means finding the points x lying in the topological closure of all the roots. Off the top of my head I guess that to identify such $x$ one might consider whether the set $\{ f(x)/f'(x) | f \in S\}$ has $0$ as a limit point, or at least something like that. $\endgroup$
    – David Feldman
    Commented Feb 21, 2012 at 23:01

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