K. Becher

activity

20122026

most citedInvolutions and stable subalgebras

9 citations · 19 across the 22 of their papers we have counts for

collaborators

25 papers

math.NT2026

Fields where torsion forms decompose

M. Archita, Karim Johannes Becher

Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum…

math.NT2026

Non- $R$ -trivial proper projective similitudes in type $A_{3} \equiv D_{3}$

M. Archita, Karim Johannes Becher

Over an arbitrary field of characteristic different from $2$ admitting an anisotropic torsion $3$ -fold Pfister form, we apply a construction due to Merkurjev to produce an algebra…

math.NT2026

Similitudes over fields with I^4=0

M. Archita, Karim Johannes Becher

This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base f…

math.RA2026

Totally decomposable algebras with involution and R-triviality

M. Archita, Karim Johannes Becher

We show that the group of proper projective similitudes of a totally decomposable algebra with orthogonal or symplectic involution of index at most 2 over a field of characteristic…

math.NT2025

Rational connectedness for groups of proper projective similitudes

M. Archita, Karim Johannes Becher

For a quadratic form $φ$ over a field of characteristic different from $2$ , we study whether its group of proper projective similitudes $PSim^{+} (φ)$ is rationally connected (i…

math.NT2025

The u-invariant of function fields in one variable

Karim Johannes Becher, Nicolas Daans, Vlerë Mehmeti

The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields…

math.NT2025

Bounds on the hermitian u-invariants under quadratic field extensions

Karim Johannes Becher, Fatma Kader Bingöl

The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants…

math.NT2024

Orthogonal involutions over fields with I^3=0

Karim Johannes Becher, Fatma Kader Bingöl

We provide upper bounds on the u-invariant for skew-hermitian forms over a quaternion algebra with its canonical involution in terms of the u-invariant of the base field F of chara…

math.NT2023

Uniform existential definitions of valuations in function fields in one variable

Karim Johannes Becher, Nicolas Daans, Philip Dittmann

We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$ . We show that…

math.NT2023

A comparison of finiteness conditions in quadratic form theory

Karim Johannes Becher, Saurabh Gosavi

We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the $u$ -invariant, the reduced stability index an…

math.AC2023★ 1 cited

A ruled residue theorem for function fields of elliptic curves

Karim Johannes Becher, Parul Gupta, Sumit Chandra Mishra

It is shown that a valuation of residue characteristic different from $2$ and $3$ on a field $E$ has at most one extension to the function field of an elliptic curve over $E$ , for…

math.RA2023★ 1 cited

Fields with bounded Brauer $2$ -torsion index

Karim Johannes Becher

It is shown that, over a field of characteristic not $2$ , the dimension of an anisotropic quadratic Pfister form of trivial total signature is at most twice the dimension of some c…

math.NT2023★ 1 cited

Sums of squares in function fields over a henselian valued field

Karim Johannes Becher, Nicolas Daans, David Grimm +2

The Pythagoras number of a field is the smallest natural number $n$ such that every sum of squares in the field is a sum of $n$ squares. Given a nondyadic henselian valued field $(…

math.RA2023

A bound on the index of exponent- $4$ algebras in terms of the $u$ -invariant

Karim Johannes Becher, Fatma Kader Bingöl

For a prime number $p$ , an integer $e \geq 2$ and a field $F$ containing a primitive $p^{e}$ -th root of unity, the index of central simple $F$ -algebras of exponent $p^{e}$ is bounded in…

math.NT2022

Bounding the Pythagoras number of a field by $2^{n} + 1$

Karim Johannes Becher, Marco Zaninelli

Given a positive integer $n$ , a sufficient condition on a field is given for bounding its Pythagoras number by $2^{n} + 1$ . The condition is satisfied for $n = 1$ by function fields of c…

math.AC2020

Square-reflexive polynomials

Karim Johannes Becher, Parul Gupta

For a field $E$ of characteristic different from $2$ and cohomological $2$ -dimension one, quadratic forms over the rational function field $E (X)$ are studied. A characterisation in…

math.RA2020★ 1 cited

The discriminant Pfister form of an algebra with involution of capacity four

Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form t…

math.NT2020★ 1 cited

Nonsplit conics in the reduction of an arithmetic curve

Karim Johannes Becher, David Grimm

For an algebraic function field $F / K$ and a discrete valuation $v$ of $K$ with perfect residue field $k$ , we bound the number of discrete valuations on $F$ extending $v$ whose resi…

math.KT2020

Strong linkage for function fields of surfaces

Karim Johannes Becher, Parul Gupta

Over a global field any finite number of central simple algebras of exponent dividing $m$ is split by a common cyclic field extension of degree $m$ . We show that the same property…

math.RA2019★ 1 cited

Splitting quaternion algebras defined over a finite field extension

Karim Johannes Becher, Fatma Kader Bingöl, David B. Leep

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a fin…

math.AC2019

A ruled residue theorem for function fields of conics

Parul Gupta, Karim Johannes Becher

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is d…

math.KT2016★ 1 cited

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_{1}$ , $Q_{2}$ over a field $F$ is not a division algebra, then there exists a separable…

math.KT2016★ 9 cited

Involutions and stable subalgebras

Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol

Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the s…

math.RA2015

Totally decomposable quadratic pairs

Karim Johannes Becher, Andrew Dolphin

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra…

Fields where torsion forms decompose

Non-R-trivial proper projective similitudes in type A3​≡D3​

Similitudes over fields with I^4=0

Totally decomposable algebras with involution and R-triviality

Rational connectedness for groups of proper projective similitudes

The u-invariant of function fields in one variable

Bounds on the hermitian u-invariants under quadratic field extensions

Orthogonal involutions over fields with I^3=0

Uniform existential definitions of valuations in function fields in one variable

A comparison of finiteness conditions in quadratic form theory

A ruled residue theorem for function fields of elliptic curves

Fields with bounded Brauer 2-torsion index

Sums of squares in function fields over a henselian valued field

A bound on the index of exponent-4 algebras in terms of the u-invariant

Bounding the Pythagoras number of a field by 2n+1

Square-reflexive polynomials

The discriminant Pfister form of an algebra with involution of capacity four

Nonsplit conics in the reduction of an arithmetic curve

Strong linkage for function fields of surfaces

Splitting quaternion algebras defined over a finite field extension

A ruled residue theorem for function fields of conics

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Involutions and stable subalgebras

Totally decomposable quadratic pairs

Non- $R$ -trivial proper projective similitudes in type $A_{3} \equiv D_{3}$

Fields with bounded Brauer $2$ -torsion index

A bound on the index of exponent- $4$ algebras in terms of the $u$ -invariant

Bounding the Pythagoras number of a field by $2^{n} + 1$