Compositions and collisions at degree p^2

Details Compositions and collisions at degree p^2 Compositions and collisions at degree p^2

2012-02-27

arxiv.org/abs/1202.5810v1

Mathematics

Commutative Algebra

Scientific paper

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. In order to count the decomposables, one has to know the number of equal-degree collisions, that is f = g o h = g^* o h^* with (g,h) != (g^*, h^*) and deg(g) = deg(g^*). 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 p^2. We provide a classification of all polynomials of degree p^2 with a collision. It yields the exact number of decomposable polynomials of degree p^2 over a finite field of characteristic p. We also present an algorithm that determines whether a given polynomial of degree p^2 has a collision or not.

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