916 research outputs found

### On rings of integers generated by their units

We give an affirmative answer to the following question by Jarden and
Narkiewicz: Is it true that every number field has a finite extension L such
that the ring of integers of L is generated by its units (as a ring)? As a part
of the proof, we generalize a theorem by Hinz on power-free values of
polynomials in number fields.Comment: 15 page

### Sums of units in function fields II - The extension problem

In 2007, Jarden and Narkiewicz raised the following question: Is it true that
each algebraic number field has a finite extension L such that the ring of
integers of L is generated by its units (as a ring)? In this article, we answer
the analogous question in the function field case.
More precisely, it is shown that for every finite non-empty set S of places
of an algebraic function field F | K over a perfect field K, there exists a
finite extension F' | F, such that the integral closure of the ring of
S-integers of F in F' is generated by its units (as a ring).Comment: 12 page

### 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

### Rational points and non-anticanonical height functions

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of
rational points of bounded height on smooth projective varieties over number
fields. We prove some new cases of this conjecture for conic bundle surfaces
equipped with some non-anticanonical height functions. As a special case, we
verify these conjectures for the first time for some smooth cubic surfaces for
height functions associated to certain ample line bundles.Comment: 16 pages; minor corrections; Proceedings of the American Mathematical
Society, 147 (2019), no. 8, 3209-322

### Arithmetic progressions in binary quadratic forms and norm forms

We prove an upper bound for the length of an arithmetic progression
represented by an irreducible integral binary quadratic form or a norm form,
which depends only on the form and the progression's common difference. For
quadratic forms, this improves significantly upon an earlier result of Dey and
Thangadurai.Comment: 7 pages; minor revision; to appear in BLM

### Forms of differing degrees over number fields

Consider a system of polynomials in many variables over the ring of integers
of a number field $K$. We prove an asymptotic formula for the number of
integral zeros of this system in homogeneously expanding boxes. As a
consequence, any smooth and geometrically integral variety $X\subseteq
\mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the
Manin-Peyre conjecture, if only its dimension is large enough compared to its
degree.
This generalizes work of Skinner, who considered the case where all
polynomials have the same degree, and recent work of Browning and Heath-Brown,
who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number
field version of the Hardy-Littlewood circle method. As a by-product, we point
out and correct an error in Skinner's treatment of the singular integral.Comment: 23 pages; minor revision; to appear in Mathematik

### Counting rational points on smooth cubic surfaces

We prove that any smooth cubic surface defined over any number field
satisfies the lower bound predicted by Manin's conjecture possibly after an
extension of small degree.Comment: 11 pages, minor revisio

### Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points
of bounded height on Fano varieties. In this paper we use conic bundles to
obtain correct lower bounds or a wide class of surfaces over number fields for
which the conjecture is still far from being proved. For example, we obtain the
conjectured lower bound of Manin's conjecture for any del Pezzo surface whose
Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after
possibly an extension of the ground field of small degree.Comment: 35 pages; final versio

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