Tame decompositions and collisions

Abstract

A univariate polynomial f over a field is decomposable if f = g • h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of F q does not divide n = deg f , is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if n has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f . In the tame case, Ritt's Second Theorem classifies all 2-collisions.We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field F q in the tame case.

Survey on Counting Special Types of Polynomials

“…They come with satisfactory (rapidly decreasing) relative error bounds except when p divides n = deg f exactly twice. Ziegler (2014) provides an exact count of tame univariate polynomials. In Section 3.4, we determine exactly the number of decomposable polynomials in one of the difficult wild cases, namely when n = p 2 .…”