Composition collisions and projective polynomials

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f = g ◦ h in Fq[x] is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p. An (equal-degree) icollision is a set of i distinct pairs (g, h) of polynomials, all with the 1 ar X iv :1 00 5. 10 87 v1 [ m at h. A C ] 6 M ay 2 01 0 same composition and deg g the same for all (g, h). Abhyankar (1997) introduced the projective polynomials xn+ax+b, where n is of the form (rm − 1)/(r − 1). Our first tool is a bijective correspondence between i-collisions of certain additive trinomials, projective polynomials with i roots, and linear spaces with i Frobenius-invariant lines. Bluher (2004b) has determined the possible number of roots of projective polynomials for m = 2, and how many polynomials there are with a prescribed number of roots. We generalize her first result to arbitrary m, and provide an alternative proof of her second result via elementary linear algebra. If one of our additive trinomials is given, we can efficiently compute the number of its decompositions, and similarly the number of roots of a projective polynomial. The runtime of these algorithms depends polynomially on the sparse input size, and thus on the input degree only logarithmically. For non-additive polynomials, we present certain decompositions and conjecture that these comprise all of the prescribed shape.