Counting decomposable polynomials with integer coefficients

TL;DR

This paper provides precise bounds and asymptotic formulas for counting decomposable polynomials with integer coefficients, focusing on fixed degree and bounded height, especially for monic polynomials of even degree.

Contribution

It introduces sharp bounds and asymptotic formulas for the enumeration of decomposable integer polynomials, advancing understanding of their distribution.

Findings

Number of monic sextic integer decomposable polynomials asymptotic to (16ζ(3)-5/4)H^3

Established sharp bounds for decomposable polynomials of fixed degree and height

Derived asymptotic formulas for even degree monic decomposable polynomials

Abstract

A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree and bounded height. Moreover, we obtain asymptotic formulas for the number of decomposable monic polynomials of even degree. For example, the number of monic sextic integer polynomials which are decomposable and of height at most $H$ is asymptotic to $(16\zeta(3)-5/4)H^3$ as $H \to \infty$.