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](http://magma.maths.usyd.edu.au/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.



Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$
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. 


