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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 q2 and every α𝔽 q × , there are infinitely many monic twin prime polynomial pairs f,f+α in 𝔽 q [T].

Theorem 2. Let f 1 (T),...,f r (T) 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 β𝔽 q for which every substitution TT l k -β with k=0,1,2,... leaves all of f 1 ,...,f r irreducible. Explicitly, the above conclusion holds provided

q2 2r 1+1 2 i=1 r degf i 2 ·

Theorem 3. Fix a finite field 𝔽 q . For each d2, define

𝒜 d :={f𝔽 q [T]:degf=dandforsomeprimelq d -1,
f(T l k )isirreduciblefork=0,1,2,...},

and let d denote the set of monic irreducibles of degree d not in 𝒜 d . Then for any ε>0,

# d q d /d 2 unconditionally,
ε q 1+εd (assumingtheabc-conjecture)·

Moreover, if we assume that

rprime,(r,q)=1 1 l q (r 2 )<,

where l q (r 2 ) denotes the multiplicative order of q modulo r 2 , then d is empty for almost all d (in the sense of asymptotic density).

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