883 research outputs found

### Counting points of fixed degree and bounded height

We consider the set of points in projective $n$-space that generate an
extension of degree $e$ over given number field $k$, and deduce an asymptotic
formula for the number of such points of absolute height at most $X$, as $X$
tends to infinity. We deduce a similar such formula with instead of the
absolute height, a so-called adelic-Lipschitz height

### Integral points of fixed degree and bounded height

By Northcott's Theorem there are only finitely many algebraic points in
affine $n$-space of fixed degree over a given number field and of height at
most $X$. For large $X$ the asymptotics of these cardinalities have been
investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In
this paper we study the case where the coordinates of the points are restricted
to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's,
Gao's and the author's results.Comment: to appear in Int. Math. Res. Notice

### Schanuel's theorem for heights defined via extension fields

Let $k$ be a number field, let $\theta$ be a nonzero algebraic number, and
let $H(\cdot)$ be the Weil height on the algebraic numbers. In response to a
question by T. Loher and D. W. Masser, we prove an asymptotic formula for the
number of $\alpha \in k$ with $H(\alpha \theta)\leq X$.
We also prove an asymptotic counting result for a new class of height
functions defined via extension fields of $k$. This provides a conceptual
framework for Loher and Masser's problem and generalizations thereof.
Moreover, we analyze the leading constant in our asymptotic formula for Loher
and Masser's problem. In particular, we prove a sharp upper bound in terms of
the classical Schanuel constant.Comment: accepted for publication by Ann. Sc. Norm. Super. Pisa Cl. Sci., 201

### Weak admissibility, primitivity, o-minimality, and Diophantine approximation

We generalise M. M. Skriganov's notion of weak admissibility for lattices to
include standard lattices occurring in Diophantine approximation and algebraic
number theory, and we prove estimates for the number of lattice points in sets
such as aligned boxes. Our result improves on Skriganov's celebrated counting
result if the box is sufficiently distorted, the lattice is not admissible,
and, e.g., symplectic or orthogonal. We establish a criterion under which our
error term is sharp, and we provide examples in dimensions $2$ and $3$ using
continued fractions. We also establish a similar counting result for primitive
lattice points, and apply the latter to the classical problem of Diophantine
approximation with primitive points as studied by Chalk, Erd\H{o}s, and others.
Finally, we use o-minimality to describe large classes of setsComment: Comments are welcom

### Small generators of function fields

Let $K/k$ be a finite extension of a global field. Such an extension can be
generated over $k$ by a single element. The aim of this article is to prove the
existence of a "small" generator in the function field case. This answers the
function field version of a question of Ruppert on small generators of number
fields

### On the Northcott property for infinite extensions

We start with a brief survey on the Northcott property for subfields of the algebraic numbers \Qbar. Then we introduce a new criterion for its validity (refining the author's previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier's theorem, stating that the maximal abelian extension of a number field $K$ contained in $K^{(d)}$ has the Northcott property, follows very easily from this refined criterion. Here $K^{(d)}$ denotes the composite field of all extensions of $K$ of degree at most $d$

### On the Northcott property and other properties related to polynomial mappings

We prove that if K/β is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/β has the Northcott property. Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km /β)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in β(d). We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewic

### On the Northcott property and other properties related to polynomial mappings

AbstractWe prove that if K/β is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/β has the Northcott property.Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/β)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in β(d).We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.</jats:p

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