Counting decomposable polynomials with integer coefficients

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 $$H \rightarrow \infty $$ .