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Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$.

Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$

of $\mathbb{Z}[x]_n$ by finite sets, i.e., a sequence of nested, finite sets $A_k$ such that $\bigcup_k A_k = \mathbb{Z}[x]_n$. (One possible sensible option is to take $A_k$ to be the set of polynomials whose coefficients are all in $\{-k, \ldots, k\}$.)

For each $A_k$ and each subgroup $H \leq S_n$ (up to isomorphism), we can ask for the proportion $p_k(H)$ of polynomials in $A_k$ whose Galois group has isomorphism type $H$.

What can be said about the limiting behavior of $p_k(H)$ as $k \to \infty$ that doesn't depend too severely on the filtration $(A_k)$? In particular, is the Galois group of a randomly selected polynomial of degree $< n$ almost surely $S_n$, or more precisely, is $$\lim_{k \to \infty} p_k(S_n) = 1?$$

If not, what is this probability? If it is $0$ or $1$, what can be said about the limiting behavior of $p_k(S_n)$?

What about these questions for other subgroups of $S_n$?

Travis Willse
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    Yes. See http://mathoverflow.net/questions/58397/the-galois-group-of-a-random-polynomial – Derek Holt Sep 20 '14 at 16:01
  • See also B. L. van der Waerden, [Die Seltenheit der Gleichungen mit Affekt][1], Mathematische Annalen 109:1 (1934), http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002276607 – Nicky Hekster Sep 20 '14 at 22:28
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    It happens with probability 1, see my question Is the Galois group associated to a random polynomial solvable with probability 0?. – Charles Sep 21 '14 at 03:15
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    Your question about lower bounds for subgroups seems like it should be very hard, since any "good" bound will prove that all finite groups can occur as Galois groups over $\mathbb{Q}$, which is still an open problem. – Tobias Kildetoft Nov 18 '14 at 09:24
  • @TobiasKildetoft One can imagine that there are interesting lower bounds for particular finite groups $H$ (say $A_n$ for polynomials of degree $n$), even if the constructions involved don't extend to general finite groups. – Travis Willse Nov 18 '14 at 09:31
  • Ahh, good point (I was somehow thinking of the question as asking for general bounds as a function of the subgroup). – Tobias Kildetoft Nov 18 '14 at 09:38
  • @TobiasKildetoft That would of course be very interesting, but like you say, it's really asking a lot. – Travis Willse Nov 18 '14 at 09:53
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    Apparently, if you take the filtration defined by size of the discriminant, instead of the size of the coefficients, the answer is rather different (https://mathoverflow.net/a/238381/56480). In this case, Malle has some conjectural lower bounds (https://www.sciencedirect.com/science/article/pii/S0022314X01927131, https://projecteuclid.org/euclid.em/1090350928), though Klüners has published a counterexample (https://projecteuclid.org/euclid.em/1090350928). – Jacob Bond Feb 17 '18 at 19:46
  • Thanks for the references, @JacobBond, very interesting stuff. – Travis Willse Feb 19 '18 at 01:10

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Following links given in the mathoverflow.se link provided by Derek Holt gives answers to some of these:

S. D. Cohen proved that the proportion of polynomials in $\mathbb{Z}[x]_n$ and coefficients all at most $k$ is absolute value for which the Galois group is not $S_n$ is $$\ll \frac{\log k}{\sqrt k},$$ so $$\lim_{k \to \infty} p_k(S_n) = 1 ,$$ and we have a lower bound on convergence.

Remark None of the sources seem, however, to give lower bounds for $p_k(H)$ for proper subgroups $H < S_n$, i.e., a description of the asymptotics of how often particular proper subgroups occur as Galois groups of polynomials in $\mathbb{Z}[x]_n$.

Travis Willse
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