Compositions and collisions at degreep2

Abstract

A univariate polynomial f over a field is decomposable if 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 formlisions 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 efficient algorithm that determines whether a given polynomial of degree p 2 has a collision or not.

Counting invariant subspaces and decompositions of additive polynomials

“…Since the foundational work of Ritt, Fatou, and Julia in the 1920s on compositions over C, a substantial body of work has been concerned with structural properties (e.g., Fried and MacRae (1969), Dorey and Whaples (1974), Schinzel (1982Schinzel ( , 2000, Zannier (1993)), with algorithmic questions (e.g., Barton and Zippel (1985), Kozen and Landau (1989)), and more recently with enumeration, exact and approximate (e.g., Giesbrecht (1988), Blankertz et al (2013), von zur Gathen (2014), Ziegler (2015Ziegler ( , 2016). A fundamental dichotomy is between the tame case, where the characteristic p of F does not divide deg g, see von zur Gathen (1990a), and the wild case, where p divides deg g, see von zur Gathen (1990b).…”