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Bhargava 2021 proves van der Waerden's conjecture about Galois groups of random integer polynomials: over all $x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ with $a_k \in \{-H, \ldots, H\}$, the number of polynomials $E_n(H)$ with Galois group not equal to the full symmetric group $S_n$ is $O(H^{n-1})$.

  • First (and this is a very basic question), am I correct that the $O$ here is in terms of $H$ only, so that in detail we have $E_n(H) < c_n H^{n-1}$ for some constants $c_n$ that depend on $n$?

  • Second, is there a conjectured growth rate for $c_n$?

From Bhargava 2021 it seems like in the worst case $c_n$ might have a factor related to the number of nonisomorphic groups of order $n$ (possibly with other factors), but presumably most of the count concentrates on fewer of the groups.

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There are two different questions you could ask here.

One is the dependence on $n$ in the conjectured asymptotic best constant, i.e. the $c_n$ such that we have $E_n (H) < c_n H^{n-1} + o(H^{n-1})$ with the little $o$ depending on $n$. This one has a precise answer, because we do not need to worry about all the groups that are proven or expected to occur with frequency $o(H^{n-1})$. So we just need to deal with the reducible polynomials, where it was computed by Chela that we can take

$$ c_n = 2^n (\zeta(n-1) - 1/2) + 2 k_n $$ where $k_n$ is the volume of the region in $\mathbb R^{n-1}$ consisting of tuples $x_1,\dots, x_{n-1}$ with $$-1 \leq x_1,\dots, x_{n-1}, \sum_{i=1}^{n-1} x_i \leq 1 .$$

It is straightforward to check that this has growth rate proportional to $2^n$ in $n$.

The other is the dependence on $n$ of the conjectured best constant valid for all $H$. This is a somewhat annoying question because it will depend heavily on small values of $H$. For example, we must take $c_n \geq 3^{n-1}$ to deal with the case where $H$ is $1$ (and the constant coefficient vanishes). It's possible that we can take $c_n \leq 3^n$ - I don't see a reason against it.

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    $\begingroup$ Thank you, that makes sense. So very roughly we might have something like $E_n(H) = O(2^n H^{n-1}) + O(1)^n o(H^{n-1})$ where all constants are absolute. Also, to clarify: do you mean "occur with frequency $o(H^{n-1})$"? $\endgroup$
    – Geoffrey Irving
    Commented Apr 22, 2022 at 21:52
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    $\begingroup$ @GeoffreyIrving Yes, that sounds exactly right, and that's what I meant to say. $\endgroup$
    – Will Sawin
    Commented Apr 22, 2022 at 22:56
  • $\begingroup$ Sorry, do you mean $k_n$ as written? Isn't it just an (n-1)-simplex of volume $2^{n-1}/(n-1)!$? $\endgroup$
    – liuyao
    Commented Apr 24, 2022 at 2:37
  • $\begingroup$ @liuyao I think it's not quite - if you take the standard $n-1$-simplex, double all coordinates, and then subtract $1$ so they range from $-1$ to $1$, then the sum of the coordinates would range from $-(n-1)$ to $-(n-3)$, not from $-1$ to $1$. $\endgroup$
    – Will Sawin
    Commented Apr 24, 2022 at 11:08
  • $\begingroup$ @WillSawin I see, it is an (n-1)-simplex still, and has fairly simple expression for the volume. The base edge isn't 2, but n-1. $\endgroup$
    – liuyao
    Commented Apr 25, 2022 at 4:01

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