Irreducible Compositions of Quadratic Polynomials have Regular Structure

A polynomial is called stable if composing it with itself an arbitrary number
of times always results in an irreducible polynomial. We expand this notion to sets
of more than one polynomial. In particular, we consider sets of monic quadratic
polynomials over a finite field, and find a criterion to decide whether all repeated
compositions of them are irreducible. Moreover, even if this is not the case, we can
describe exactly which compositions are irreducible by showing that they have the
structure of a regular language.