Compositions and collisions at degree p^2

TL;DR

This paper classifies all degree p^2 polynomials with collisions over finite fields of characteristic p, providing exact counts and an efficient algorithm to detect such collisions, advancing understanding of polynomial decompositions in the wild case.

Contribution

It offers a complete classification of degree p^2 polynomials with collisions and an algorithm to determine collision presence, improving bounds in the wild case.

Findings

Exact count of decomposable degree p^2 polynomials over finite fields.

Complete classification of polynomials with collisions at degree p^2.

Efficient algorithm to detect collisions in degree p^2 polynomials.

Abstract

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 wants to know, under a suitable normalization, the number of equal-degree collisions of the form 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 efficient algorithm that determines whether a given polynomial of degree p^2 has a collision or not.