arXiv subjects

Karim Johannes Becher

Publications and source records attributed to Karim Johannes Becher.

At least 19 recordsLinked to original sources

arXiv2026

Sums of squares in function fields over a henselian valued field

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 $(K, v)$, we show that the best bound on the Pythagoras number for function fields of curves over $K$ is essentially the same as for function fields of curves over the residue field $Kv$. As a consequence, we obtain the upper bound $5$ for the Pythagoras number of function fields of curves over any field $K$ for which the rational function field $K(X)$ has Pythagoras number $2$. The proof uses a reduction to the case where the value group is a subgroup of the real numbers, where then a local-global principle for quadratic forms due to V.~Mehmeti is applied.

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arXiv2026

Totally decomposable algebras with involution and R-triviality

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 different from 2 is R-trivial. We further give an example showing that the statement does not extend to index 4 in the orthogonal case.

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arXiv2026

Fields where torsion forms decompose

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 of 2-dimensional torsion forms. This is obtained from a more general study of weakly isotropic forms over henselian valued fields and over function fields in one variable.

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arXiv2026

Non-$R$-trivial proper projective similitudes in type $A_3\equiv D_3$

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 with orthogonal involution of degree $6$ which admits proper projective similitudes that are not $R$-trivial. In particular, such examples exist over every finitely generated transcendental extension of a local or global number field, as well as over every finitely generated extension of transcendence degree $3$ of $\mathbb{R}$.

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arXiv2026

Similitudes over fields with I^4=0

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 fields of characteristic different from 2 over which every 9-dimensional quadratic form has a nontrivial zero. This includes function fields of p-adic curves and extensions of transcendence degree 3 of C. Main results of [28] and [29] are extended by relaxing the condition on the base field as well as on the Clifford invariant for orthogonal involutions.

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arXiv2025

Uniform existential definitions of valuations in function fields in one variable

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, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and finitely generated integrally closed $K$-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

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arXiv2025

The u-invariant of function fields in one variable

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 in one variable over a henselian valued field with arbitrary value group and with residue field of characteristic different from 2. This generalises a theorem due to Harbater, Hartmann and Krashen and its extension due to Scheiderer. Their result covers the special case where the valuation is discrete. We further give a new proof of a theorem due to Parimala and Suresh bounding by 8 the u-invariant of a function field in one variable over any henselian discretely valued field of characteristic 0 with perfect residue field of characteristic 2.

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arXiv2025

Rational connectedness for groups of proper projective similitudes

For a quadratic form $φ$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\bf PSim}^+(φ)$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient conditions in terms of structure properties of $φ$. We further provide new examples of quadratic forms $φ$ belonging to a given power of the fundamental ideal in the Witt ring and such that ${\bf PSim}^+(φ)$ is not rationally connected.

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arXiv2025

Orthogonal involutions over fields with I^3=0

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 characteristic different from 2 when I^3F = 0.

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arXiv2025

Bounds on the hermitian u-invariants under quadratic field extensions

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 under a quadratic field extension. This is applied to obtain bounds in terms of the index and the u-invariant of the base field.

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arXiv2024

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

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 that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree $8$ algebras in characteristic 2.

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arXiv2024

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

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 curves over iterated formal power series fields over $\mathbb{R}$, as well as by finite field extensions of $\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$ on the Pythagoras number. The new method presented here might help to establish more generally $2^n+1$ as an upper bound for the Pythagoras number of function fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$.

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arXiv2023

Nonsplit conics in the reduction of an arithmetic curve

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 residue fields are algebraic function fields of genus zero over $k$ but not ruled. Assuming that $K$ is relatively algebraically closed in $F$, we find that the number of nonruled residually transcendental extensions of $v$ to $F$ is bounded by $\mathfrak{g}+1$ where $\mathfrak{g}$ is the genus of $F/K$. An application to sums of squares in function fields of curves over $\mathbb{R}(\!(t)\!)$ is presented.

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arXiv2023

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

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 terms of the $p$-symbol length of $F$. For a nonreal field $F$ of characteristic different from $2$, the index of central simple algebras of exponent $4$ is bounded in terms of the $u$-invariant of $F$. Finally, a new construction for nonreal fields of $u$-invariant $6$ is presented.

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arXiv2023

A ruled residue theorem for function fields of elliptic curves

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 which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.

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arXiv2023

A comparison of finiteness conditions in quadratic form theory

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 and the symbol lengths for Galois cohomology groups with coefficients in $μ_2=\{+1,-1\}$, as well as a new invariant called the splitting height.

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arXiv2023

Fields with bounded Brauer $2$-torsion index

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 central division algebra of exponent $2$. The proof is based on computations with quadratic trace forms of central simple algebras.

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arXiv2021

Square-reflexive polynomials

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 terms of polynomials in $E[X]$ is obtained for having that quadratic forms over $E(X)$ satisfy a local-global principle with respect to discrete valuations that are trivial on $E$. In this way new elementary proofs for the local-global principle are achieved in the cases where $E$ is finite or pseudo-algebraically closed. The study is complemented by various examples.

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