On the proportion of irreducible polynomials in unicritically generated semigroups

Abstract

Let p be a prime number and let S = {x^p+c₁,…,x^p+c_r} be a finite set of unicritical polynomials for some c₁,…,c_r ∈ Z. Moreover, assume that S contains at least one irreducible polynomial over Q. Then we construct a large, explicit subset of irreducible polynomials within the semigroup generated by S under composition; in fact, we show that this subset has positive asymptotic density within the full semigroup when we count polynomials by degree. In addition, when p = 2 we construct an infinite family of semigroups that break the local-global principle for irreducibility. To do this, we use a mix of algebraic and arithmetic techniques and results, including Runge's method, the elliptic curve Chabauty method, and Fermat's Last Theorem.

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