66 research outputs found
The set of stable primes for polynomial sequences with large Galois group
Let be a number field with ring of integers , and let
be a sequence of monic
polynomials such that for every , the composition
is irreducible. In this paper we
show that if the size of the Galois group of is large enough (in a
precise sense) as a function of , then the set of primes such that every is irreducible modulo
has density zero. Moreover, we prove that the subset of
polynomial sequences such that the Galois group of is large enough
has density 1, in an appropriate sense, within the set of all polynomial
sequences.Comment: Comments are welcome
An equivariant isomorphism theorem for mod reductions of arboreal Galois representations
Let be a quadratic, monic polynomial with coefficients in , where is a localization of a number ring
. In this paper, we first prove that if is non-square and
non-isotrivial, then there exists an absolute, effective constant with
the following property: for all primes
such that the reduced polynomial is non-square and non-isotrivial, the squarefree
Zsigmondy set of is bounded by . Using this
result, we prove that if is non-isotrivial and geometrically stable then
outside a finite, effective set of primes of the geometric
part of the arboreal representation of is isomorphic to
that of . As an application of our results we prove R. Jones' conjecture
on the arboreal Galois representation attached to the polynomial .Comment: Comments are welcome
On Mertens-Ces\`aro Theorem for Number Fields
Let be a number field with ring of integers . After
introducing a suitable notion of density for subsets of ,
generalizing that of natural density for subsets of , we show that
the density of the set of coprime -tuples of algebraic integers is
, where is the Dedekind zeta function of .Comment: To appear in the Bulletin of the Australian Mathematical Societ
Irreducible compositions of degree two polynomials over finite fields have regular structure
Let be an odd prime power and be the set of monic irreducible
polynomials in which can be written as a composition of monic
degree two polynomials. In this paper we prove that has a natural regular
structure by showing that there exists a finite automaton having as
accepted language. Our method is constructive.Comment: To appear in The Quarterly Journal of Mathematic
Exceptional scatteredness in prime degree
Let be an odd prime power and be a positive integer. Let be a -linearised -scattered polynomial of linearized
degree . Let be an odd prime number. In this paper we show
that under these assumptions it follows that . Our technique involves a
Galois theoretical characterization of -scattered polynomials combined with
the classification of transitive subgroups of the general linear group over the
finite field
Constraining images of quadratic arboreal representations
In this paper, we prove several results on finitely generated dynamical
Galois groups attached to quadratic polynomials. First we show that, over
global fields, quadratic post-critically finite polynomials are precisely those
having an arboreal representation whose image is topologically finitely
generated. To obtain this result, we also prove the quadratic case of Hindes'
conjecture on dynamical non-isotriviality. Next, we give two applications of
this result. On the one hand, we prove that quadratic polynomials over global
fields with abelian dynamical Galois group are necessarily post-critically
finite, and we combine our results with local class field theory to classify
quadratic pairs over with abelian dynamical Galois group, improving
on recent results of Andrews and Petsche. On the other hand we show that
several infinite families of subgroups of the automorphism group of the
infinite binary tree cannot appear as images of arboreal representations of
quadratic polynomials over number fields, yielding unconditional evidence
towards Jones' finite index conjecture.Comment: Sections 3 and 4 now swapped. Accepted for publication on IMR
Number Theoretical Locally Recoverable Codes
In this paper we give constructions for infinite sequences of finite
non-linear locally recoverable codes over a product of finite fields arising
from basis expansions in algebraic number fields. The codes in our sequences
have increasing length and size, constant rate, fixed locality, and minimum
distance going to infinity
On the logarithmic probability that a random integral ideal is -free
This extends a theorem of Davenport and Erd\"os on sequences of rational
integers to sequences of integral ideals in arbitrary number fields . More
precisely, we introduce a logarithmic density for sets of integral ideals in
and provide a formula for the logarithmic density of the set of so-called
-free ideals, i.e. integral ideals that are not multiples of any
ideal from a fixed set .Comment: 9 pages, to appear in S. Ferenczi, J. Ku{\l}aga-Przymus and M.
Lema\'nczyk (eds.), Chowla's conjecture: from the Liouville function to the
M\"obius function, Lecture Notes in Math., Springe
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