Compositions and collisions at degree p2p2

A univariate polynomial f   over a field is decomposable if f=g∘h=g(h)f=g∘h=g(h) for nonlinear polynomials g and h  . In order to count the decomposables, one wants to know, under a suitable normalization, the number of equal-degree collisions of the form f=g∘h=g⁎∘h⁎f=g∘h=g⁎∘h⁎ with (g,h)≠(g⁎,h⁎)(g,h)≠(g⁎,h⁎) and degg=degg⁎. Such collisions only occur in the wild case, where the field characteristic p divides deg f  . Reasonable bounds on the number of decomposables over a finite field are known, but they are less sharp in the wild case, in particular for degree p2p2.We provide a classification of all polynomials of degree p2p2 with a collision. It yields the exact number of decomposable polynomials of degree p2p2 over a finite field of characteristic p  . We also present an efficient algorithm that determines whether a given polynomial of degree p2p2 has a collision or not.