[Bhargava 2021](https://arxiv.org/abs/2111.06507) 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](https://arxiv.org/abs/2111.06507) 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.



Is there a conjectured dependence on $n$ in van der Waerden's conjecture?
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.


