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arXiv · 2308.14202

On the proportion of irreducible polynomials in unicritically generated semigroups

Wade Hindes·Reiyah Jacobs·Benjamin Keller·Albert Kim·Peter Ye·Aaron Zhou
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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.

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Wade Hindes, Reiyah Jacobs, Benjamin Keller, Albert Kim, Peter Ye, Aaron Zhou. 2023-08-27. On the proportion of irreducible polynomials in unicritically generated semigroups. https://arxiv.org/abs/2308.14202

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