Tame Decompositions and Collisions

TL;DR

This paper develops a normal form and an efficient algorithm to precisely count decomposable polynomials and collisions in polynomial decompositions over finite fields, especially in the tame case.

Contribution

It introduces a normal form for multi-collisions and provides an exact, computable formula for counting collisions and decomposables in the tame case.

Findings

Derived an exact formula for the number of collisions of polynomial decompositions.

Provided an algorithm to compute the number of decomposable polynomials of degree n.

Extended understanding of polynomial decompositions beyond the two-prime-factor case.

Abstract

A univariate polynomial f over a field is decomposable if f = g o 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 Fq 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 Fq in the tame case.