A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Higher-degree Artin conjecture
Tekijät: Järviniemi Olli
Kustantaja: Oxford University Press
Julkaisuvuosi: 2024
Journal: Quarterly Journal of Mathematics
Tietokannassa oleva lehden nimi: The Quarterly Journal of Mathematics
Artikkelin numero: haae012
Vuosikerta: 75
Numero: 2
Aloitussivu: 521
Lopetussivu: 547
ISSN: 0033-5606
eISSN: 1464-3847
DOI: https://doi.org/10.1093/qmath/haae012
Verkko-osoite: https://doi.org/10.1093/qmath/haae012
Tiivistelmä
For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.
For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.
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