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An explicit approach to hypothesis H for polynomials over a finite field. (English) Zbl 1187.11046
De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 259-273 (2008).
The author proves: Theorem 1. For every $q\neq 2$ and every $\alpha\in\mathbb{F}_q^\times$, there are infinitely many monic twin prime polynomial pairs $f,f+\alpha$ in $\mathbb{F}_q[T]$. Theorem 2. Let $f_1(T),\ldots,f_r(T)$ be irreducible polynomials over $\mathbb{F}_q$. If $q$ is large compared to both $r$ and the sum of the degrees of the $f_i$, then there is a prime $l$ dividing $q-1$ and an element $\beta\in\mathbb{F}_q$ for which every substitution $T\to T^{l^k}-\beta$ with $k=0,1,2,\ldots$ leaves all of $f_1,\ldots,f_r$ irreducible. Explicitly, the above conclusion holds provided $$q\geq 2^{2r}\left(1+\frac12\sum_{i=1}^r\deg f_i\right)^2.$$ Theorem 3. Fix a finite field $\mathbb{F}_q$. For each $d\geq 2$, define $$\mathcal{A}_d:=\{f\in\mathbb{F}_q[T]:\deg f=d\,\text{and for some prime}\,l\mid q^d-1,$$ $$f(T^{l^k})\,\text{is irreducible for}\,k=0,1,2,\ldots\},$$ and let $\mathcal{E}_d$ denote the set of monic irreducibles of degree $d$ not in $\mathcal{A}_d$. Then for any $\varepsilon>0,$ $$\#\mathcal{E}_d\ll q^d/d^2\quad\text{unconditionally},$$ $$\ll_{\varepsilon}q^{1+\varepsilon d}\quad\text{(assuming the abc-conjecture)}.$$ Moreover, if we assume that $$\sum_{r\,\text{prime},(r,q)=1}\frac{1}{l_q(r^2)}<\infty,$$ where $l_q(r^2)$ denotes the multiplicative order of $q$ modulo $r^2$, then $\mathcal{E}_d$ is empty for almost all $d$ (in the sense of asymptotic density). For the entire collection see [Zbl 1142.11002].

11T55Arithmetic theory of polynomial rings over finite fields
11N32Primes represented by polynomials