Gleason's polynomials are the sequence of monic integer polynomials defined recursively by $$ \prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}], $$ for $n=1,2, \ldots$. Thus they start out like: $$ \begin{align} G_1(c) &= c, \\ G_2(c) &= c+1, \\ G_3(c) &= c^3 + 2c^2 + c + 1, \\ G_4(c) &= c^6 + 3c^5 + 3c^4 + 3c^3 + 2c^2 + 1 \end{align} $$ They give the period-$n$ centers for the hyperbolic components of the Mandelbrot set in complex dynamics. In many ways they resemble the cyclotomic polynomials, which would result if we had $c^n-1$ on the right-hand side of the recursive definition; or the dynatomic polynomials in the dynamical plane (this is their version in the parameter plane parametrizing the quadratic iterations $z^2+c$).
For example, $\mathrm{Res}(G_n,G_m) = \pm 1$ for any pair $n \neq m$, just as for the cyclotomic polynomials. This is proved for instance in Corollary 4.8 of this paper by Hutz and Towsley. (Courtesy of Matt Baker for this reference. An aside: What are the exact signs?)
My question is about the discriminants of these polynomials:
What are the lower and upper growth rates of $\delta_n := \log{|\mathrm{Disc}(G_n)|}$? Does $\frac{1}{n}\sum_{d \mid n} \delta_d \asymp \log{n}$?
(The latter $\sim \log{n}$ for the cyclotomics, and the upper and lower growth rates $\sim \phi(n)\log{n}$ for the individual cyclotomic discriminants are $n\log{n}$ and $e^{-\gamma}n\frac{\log{n}}{\log{\log{n}}}$, by Mertens's theorem.)
I was wondering in what ways would the Gleason discriminants behave similarly to the cyclotomic discriminants (size-wise?), and in what ways they are markedly different. One marked difference is that the prime factors of the $\mathrm{Disc}(G_n)$ are quite unpredictable; a casual look at the first few prime factorizations of the discriminants of $G_3, G_4, G_5$ and $G_6$ reveals $$ 23 \times 2551, \quad 13 \times 24554691821639909, \quad 13^2 \times 949818439 \times 6488190752068386528993226361, \quad 8291 × 9137 × 420221 × 189946 395389 × 4813 162343 551332 730513 × 2 837919 018511 214750 008829 × 1 858730 157152 877176 856713 108209 153714 699601 $$
The one thing that is easy and very useful to see is that these discriminants are odd: this is how Gleason established that the complex roots of the polynomials are all distinct (no multiplicities), which is non-obvious from the definition. We certainly have the trivial lower $\delta_n \gg n$ (from Minkowski) and upper $\delta_n = o(n^2)$ (from noting that the Mandelbrot set has logarithmic capacity $1$) estimates, but both of these are on purely general grounds, and they leave out a large margin.
