On the proportion of irreducible polynomials in unicritically generated semigroups

TL;DR

This paper studies the distribution of irreducible polynomials within semigroups generated by unicritical polynomials, providing explicit large subsets with positive density and constructing examples that violate local-global principles for irreducibility.

Contribution

It constructs explicit large subsets of irreducible polynomials in semigroups generated by unicritical polynomials and demonstrates violations of local-global irreducibility principles for specific primes.

Findings

Large explicit subsets of irreducible polynomials with positive density

Construction of semigroups breaking local-global irreducibility principles for p=2,3

Application of algebraic and arithmetic techniques like Runge's method and elliptic curve Chabauty

Abstract

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. 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 $p=2$ or $3$ 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.