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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 $q\ne 2$ and every $\alpha \in {𝔽}_{q}^{×}$, there are infinitely many monic twin prime polynomial pairs $f,f+\alpha$ in ${𝔽}_{q}\left[T\right]$.

Theorem 2. Let ${f}_{1}\left(T\right),...,{f}_{r}\left(T\right)$ be irreducible polynomials over ${𝔽}_{q}$. 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 $\beta \in {𝔽}_{q}$ for which every substitution $T\to {T}^{{l}^{k}}-\beta$ with $k=0,1,2,...$ leaves all of ${f}_{1},...,{f}_{r}$ irreducible. Explicitly, the above conclusion holds provided

$q\ge {2}^{2r}{\left(1+\frac{1}{2}\sum _{i=1}^{r}deg{f}_{i}\right)}^{2}·$

Theorem 3. Fix a finite field ${𝔽}_{q}$. For each $d\ge 2$, define

${𝒜}_{d}:=\left\{f\in {𝔽}_{q}\left[T\right]:degf=d\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\text{prime}\phantom{\rule{0.166667em}{0ex}}l\mid {q}^{d}-1,$
$f\left({T}^{{l}^{k}}\right)\phantom{\rule{0.166667em}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{irreducible}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{0.166667em}{0ex}}k=0,1,2,...\right\},$

and let ${ℰ}_{d}$ denote the set of monic irreducibles of degree $d$ not in ${𝒜}_{d}$. Then for any $\epsilon >0,$

$#{ℰ}_{d}\ll {q}^{d}/{d}^{2}\phantom{\rule{1.em}{0ex}}\text{unconditionally},$
${\ll }_{\epsilon }{q}^{1+\epsilon d}\phantom{\rule{1.em}{0ex}}\text{(assuming}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{abc-conjecture)}·$

Moreover, if we assume that

$\sum _{r\phantom{\rule{0.166667em}{0ex}}\text{prime},\left(r,q\right)=1}\frac{1}{{l}_{q}\left({r}^{2}\right)}<\infty ,$

where ${l}_{q}\left({r}^{2}\right)$ denotes the multiplicative order of $q$ modulo ${r}^{2}$, then ${ℰ}_{d}$ is empty for almost all $d$ (in the sense of asymptotic density).

##### MSC:
 11T55 Arithmetic theory of polynomial rings over finite fields 11N32 Primes represented by polynomials
##### Keywords:
finite fields; irreducible polynomials