Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results done by Magma, there are only 24 kinds of factorizations of different degrees appearing when I factor 100 random such polynomials in $F_{81}[x]$ and factors of degrees {5, 20, 56} appear 5 times, factors of degrees {1, 1, 2, 3, 6, 11, 25, 32} appear 5 times, factors of degrees {1, 1, 2, 3, 8 ,66} appear 8 times, … . What is the cause of this phenomenon? There are papers concerning the factorization of $f(x)=x^q-(bx+c)\in \mathbb{F}_q[x]$ like http://dml.cz/dmlcz/126360, but I cannot find a suitable one for my type.
 A: Actually the factorization patterns don't look all that special:
they feel like the cycle structures of random permutations in
the symmetric group $S_{81}$, unlike the special case $a=0$,
and unlike the (somewhat more interesting) polynomials $x^{q+1}+bx+c$.
The repetitions do require explanation, but the explanation is simply that
the polynomials themselves are repeating: they seem to vary over $q^3$ choices
of $(a,b,c)$, but it's only about $q$ up to translation and scaling.
If $a \neq 0$ then $f(x - b/2a)$ yields a polynomial of the same form 
and with the same $a$, but with $b=0$; and then changing $f(x)$ to
$af(x/a)$ yields an equivalent polynomial with $a=1$ and $b=0$.
When $q = p^f$ with $f>1$ there are also field automorphisms:
$x^q-x-c$ is equivalent to $x^q-x-c^{p^e}$ for each $e=1,2,\ldots,f-1$.
So for $q=81=3^4$ there are only about $q/f \approx 20$ distinct possibilities, 
and in 100 tries you expect to see each pattern appear about $100/20=5$ 
times on average, just as you describe.
A: It is a special case of thm 2.3 in http://arxiv.org/abs/1302.0625  (version 3, I've just updated it, it has a different number in version 2) that the factorization pattern of polynomials $x^q-(ax^2+bx+c)$ distributes the same as random permutation in $S_n$ (unless $q$ is a power of $2$). In fact this theorem deals with $h(x) + (ax^2+bx+c)$, uniformly on $h$ of fixed degrees.
In the case $x^{p^2} - (bx+c)$, for example, the factorization is related to cycle structure of permutations in the group of transformations of the affine line over $\mathbb{F}_{p^2}$, while in $x^{p^2+1}-(bx+c)$ it is related to the transformations of the projective line. See last section in the ms. 
