Counting decomposable univariate polynomials

TL;DR

This paper investigates the structure and enumeration of decomposable univariate polynomials over different fields, providing new dimension results and approximation bounds, especially distinguishing between tame and wild cases.

Contribution

It determines the dimension of the set of decomposable polynomials over algebraically closed fields and offers approximations for their count over finite fields, with improved bounds in the tame case.

Findings

Dimension of decomposable polynomials over algebraically closed fields is established.

Approximate count of decomposable polynomials over finite fields with error bounds.

Exponential error bounds in the tame case, weaker bounds in the wild case.

Abstract

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.