Generating integer polynomials from X ² and X ³ using function composition: a study of subnearrings of (ℤ[X], +, ◦)

Abstract

Which integer polynomials can we write down if the only exponent to be used is 3? Such problems can be considered as instances of the subnearring generation problem. We show that the nearring (ℤ[x], +, ◦) of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of {1, x, x ², x ³ }.