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.
Here's the link at MO for the same question.
Answered on mathoverflow: the individual degree patterns don't have much structure, and the repetitions occur because over a field of $q=p^f$ elements there are only about $q/f$ distinct choies of $f$ after accounting for translation, scaling, and field automorphism (so for example $q=81$ yields about $20$ so we expect each pattern to appear about $100/20=5$ times as observed).