arXiv subjects

Nico Lorenz

Publications and source records attributed to Nico Lorenz.

9 recordsLinked to original sources

arXiv2026

Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

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arXiv2026

A cohomological invariant for algebras of degree 8 and exponent 2 in characteristic 2

Our aim in this paper is to extend a work of Sivatski to characteristic 2. More precisely, for $F$ a field of characteristic $2$ and a central simple algebra $A$ of exponent 2 that splits over a triquadratic extension of $F$ of separability degree at least 4, we attach a cohomological invariant $\inv(A) \in H_2^3(F) / G$, where $H_2^3(F)$ is the third Kato-Milno cohomology group and $G$ is a subgroup of $H_2^3(F)$ divisible by the Brauer class of $A$. As an application, we will relate the decomposability of the algebra in degree 8 to the vanishing of $\inv(A)$. Moreover, we will use this invariant to prove some descent results for central simple algebras and quadratic forms over biquadratic extensions.

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arXiv2025

Diameter and Girth of Representation Graphs of Quadratic Forms

Let $q$ be a non-degenerate quadratic form defined on an $F$ vector space $V$ and $a \in F$. We consider the Cayley graph on $V$ with generating set $\{x \in V \mid q(x) = a\}$ and study its diameter and girth. In particular, if $F$ is a finite field, we calculate these invariants and the number of cycles of minimal length in these graphs.

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arXiv2024

Isotropy indices of Pfister multiples in characteristic two

Let $F$ be a field of characteristic $2$, $π$ be an $n$-fold bilinear Pfister form over $F$ and $φ$ an arbitrary quadratic form over $F$. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy indices of $φ$ and $π\otimesφ$ and prove relations among the indices of these two forms over certain field extensions.

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arXiv2024

On the Symbol Length of Fields with finite Square Class Number

Let $F$ be a field of characteristic not $2$ with finitely many square classes. Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over $F$. Moreover, we compute upper bounds for the $n$-symbol length $F$ ($n\in\mathbb N$), i.e., the smallest integer $\mathrm{sl}_n(F)\geq 0$ such that to each quadratic form $ϕ\in \mathsf I^n(F)$ there exists some $0\leq k\leq \mathrm{sl}_n(F)$ and Pfister forms $π_1,\ldots, π_k$ such that $φ\equiv π_1+\ldots+π_k\mod \mathsf I^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.

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arXiv2024

On Generalised Albert Forms over Discretely Valued Fields

For a discrete valuation ring $R$ with quotient field $K$ and residue field $F$ both of characteristic not 2, we study low-dimensional quadratic forms with Witt class in the $n$-th power of the fundamental ideal of $F$ resp. $K$ and point out connections between forms over these fields. We analyse the minimal number of Pfister forms such that a given form is Witt equivalent to the sum of these and study forms congruent modulo a higher power of the fundamental ideal towards similarity.

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arXiv2023

Cliques in Representation Graphs of Quadratic Forms

We study cliques in graphs arising from quadratic forms where the vertices are the elements of the module of the quadratic form and two vertices are adjacent if their difference represents some fixed scalar. We determine structural properties and the clique number for quadratic forms over finite rings. We further extend previous results about graphs arising from such forms and forms over fields of characteristic 0 in a unified framework.

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arXiv2021

Pfister Numbers over Rigid Fields

For certain types of quadratic forms lying in the n-th power of the fundamental ideal, we compute upper bounds and where possible exact values for the minimal number of general n-fold Pfister forms, that are needed to write the Witt class of that given form as the sum of the Witt classes of those n-fold Pfister forms. We restrict ourselves mostly to the case of so called rigid fields, i.e. fields in which binary anisotropic forms represent at most 2 square classes.

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arXiv2020

Supreme Torsion Forms

We study formally real, non-pythagorean fields which have an anisotropic torsion form that contains every anisotropic torsion form as a subform. We obtain consequences for certain invariants and the Witt ring of such fields and construct examples. We obtain a theory analogous to the theory of supreme Pfister forms introduced by Karim Becher and see examples in which the Pythagoras number for formally real fields behaves like the level for nonreal fields.

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