Martin Widmer

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20112026

most citedCounting primitive points of bounded height

27 citations · 120 across the 16 of their papers we have counts for

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23 papers · 1 filter

math.NT2026

Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density

Angelot Behajaina, Pierre Dèbes, Martin Widmer

The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart whe…

math.NT2026

Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields

Christopher Frei, Efthymios Sofos, Martin Widmer

We prove an effective version of the Chebotarev density theorem, conditional on the quasi-Riemann hypothesis for the Dedekind zeta function. Drawing on recent breakthroughs of Open…

math.NT2023

On Mahler's inequality and small integral generators of totally complex number fields

Murray Child, Martin Widmer

We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $d \geq 2$ su…

math.NT2023

Small integral generators of totally complex number fields

Shabnam Akhtari, Jeffrey Vaaler, Martin Widmer

Let $K$ be an algebraic number field and $H$ the absolute Weil height. Write $c_{K}$ for a certain positive constant that is an invariant of $K$ . We consider the question: does $K$ c…

math.NT2021

Northcott numbers for the house and the Weil height

Fabien Pazuki, Niclas Technau, Martin Widmer

For an algebraic number $α$ and $γ \in R$ , $h (α)$ be the (logarithmic) Weil height, and $h_{γ} (α) = (deg α)^{γ} h (α)$ be the $γ$ -weighted (logarithmic) Weil height of $α$ .…

math.NT2020

Lehmer without Bogomolov

Fabien Pazuki, Niclas Technau, Martin Widmer

We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.

math.NT2020

Bertini and Northcott

Fabien Pazuki, Martin Widmer

We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general…

math.NT2018

Averages and higher moments for the $ℓ$ -torsion in class groups

Christopher Frei, Martin Widmer

We prove upper bounds for the average size of the $ℓ$ -torsion $Cl_{K} [ℓ]$ of the class group of $K$ , as $K$ runs through certain natural families of number fields and $\…

math.NT2017★ 18 cited

Bounds for the $ℓ$ -torsion in class groups

Martin Widmer

We prove for each integer $ℓ \geq 1$ an unconditional upper bound for the size of the $ℓ$ -torsion subgroup $C l_{K} [ℓ]$ of the class group of $K$ , which holds for all but a ze…

math.NT2017★ 6 cited

Average bounds for the $ℓ$ -torsion in class groups of cyclic extensions

Christopher Frei, Martin Widmer

For all positive integers $ℓ$ , we prove non-trivial bounds for the $ℓ$ -torsion in the class group of $K$ , which hold for almost all number fields $K$ in certain families of c…

math.NT2016★ 3 cited

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

Martin Widmer

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 pr…

math.NT2016★ 4 cited

On a Counting Theorem of Skriganov

Niclas Technau, Martin Widmer

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box, which generalises a cou…

math.NT2014

Asymptotic Diophantine approximation: The multiplicative case

Martin Widmer

Let $α$ and $β$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q \leq Q$ that satisfy $\|qα\|\cdot\|qβ\|<\F$. If we cho…

math.NT2013

Integral points of fixed degree and bounded height

Martin Widmer

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 asym…

math.NT2012

Counting lattice points and o-minimal structures

Fabrizio Barroero, Martin Widmer

Let $Λ$ be a lattice in $R^{n}$ , and let $Z \subseteq R^{m + n}$ be a definable family in an o-minimal structure over $R$ . We give sharp estimates for the number of lattice points in…

math.NT2012

Schanuel's theorem for heights defined via extension fields

Christopher Frei, Martin Widmer

Let $k$ be a number field, let $θ$ 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. Ma…

math.NT2012★ 23 cited

On certain infinite extensions of the rationals with Northcott property

Martin Widmer

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states…

math.NT2012★ 4 cited

Small generators of function fields

Martin Widmer

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" g…

math.NT2012★ 3 cited

On number fields with nontrivial subfields

Martin Widmer

What is the probability for a number field of composite degree $d$ to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of pro…

math.NT2012★ 24 cited

Counting points of fixed degree and bounded height

Martin Widmer

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…

math.NT2012★ 27 cited

Counting primitive points of bounded height

Martin Widmer

Let $k$ be a number field and $K$ a finite extension of $k$ . We count points of bounded height in projective space over the field $K$ generating the extension $K / k$ . As the height…

math.NT2012

A note on generators of number fields

Jeffrey D. Vaaler, Martin Widmer

We establish upper bounds for the smallest height of a generator of a number field $k$ over the rational field $\Q$. Our first bound applies to all number fields $k$ having at leas…

math.NT2011★ 8 cited

Counting points of fixed degree and given height over function fields

Jeffrey Lin Thunder, Martin Widmer

Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $k ˉ$ its algebraic closure. We count points in projective space $P^{n - 1} (k ˉ)$ with given h…

Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density

Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields

On Mahler's inequality and small integral generators of totally complex number fields

Small integral generators of totally complex number fields

Northcott numbers for the house and the Weil height

Lehmer without Bogomolov

Bertini and Northcott

Averages and higher moments for the ℓ-torsion in class groups

Bounds for the ℓ-torsion in class groups

Average bounds for the ℓ-torsion in class groups of cyclic extensions

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

On a Counting Theorem of Skriganov

Asymptotic Diophantine approximation: The multiplicative case

Integral points of fixed degree and bounded height

Counting lattice points and o-minimal structures

Schanuel's theorem for heights defined via extension fields

On certain infinite extensions of the rationals with Northcott property

Small generators of function fields

On number fields with nontrivial subfields

Counting points of fixed degree and bounded height

Counting primitive points of bounded height

A note on generators of number fields

Counting points of fixed degree and given height over function fields

Averages and higher moments for the $ℓ$ -torsion in class groups

Bounds for the $ℓ$ -torsion in class groups

Average bounds for the $ℓ$ -torsion in class groups of cyclic extensions