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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].

MSC:

11T55 Arithmetic theory of polynomial rings over finite fields
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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