arXiv subjects

Detlev W. Hoffmann

Publications and source records attributed to Detlev W. Hoffmann.

7 recordsLinked to original sources

arXiv2026

Nonsimilar half-neighbors over fields of characteristic 2

The total isotropy index of a quadratic form $\varphi$ over a field $F$ is the maximum dimension of any totally isotropic subspace of $\varphi$. If $\varphi$ is anisotropic and $\psi$ is another anisotropic quadratic form over $F$ of the same dimension, then $\varphi$ and $\psi$ are called Vishik-equivalent if, over any field extension $E/F$, their total isotropy indices are the same. In characteristic $\neq 2$, Vishik-equivalence implies similarity in all dimensions $\leq 7$ and in all odd dimensions, but there are counterexamples in all even dimensions $\geq 8$. In this paper, we construct semi-singular anisotropic quadratic forms of dimension $2^m$ for any $m\geq 3$ and defined over a suitable extension of any given field $F_0$ of characteristic $2$ that are Vishik-equivalent but not similar, thus completing the list of such examples provided earlier by the first author and Krist\'yna Zemkov\'a.

math.NT↗

arXiv2020

Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2

We give examples of quaternion and octonion division algebras over a field $F$ of characteristic $2$ that split over a purely inseparable extension $E$ of $F$ of degree $\geq 4$ but that do not split over any subextension of $F$ inside $E$ of lower exponent, or, in the case of octonions, over any simple subextension of $F$ inside $E$. Thus, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms.

math.RA↗

arXiv2020

Similarity of quadratic and symmetric bilinear forms in characteristic 2

We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic $2$. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic $2$, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic $2$.

math.NT↗

arXiv2019

Sums of integers and sums of their squares

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear forms. Based on this approach, we characterize $\mathscr{S}_m(n)$ for all $m\leq 11$ and provide a few partial results for arbitrary $m$. We also show how Mordell's results can be used to study variations of the original problem where the sum of the integers is replaced by a linear form in these integers. In this way, we recover and generalize earlier results by Z.W. Sun et. al..

math.NT↗

arXiv2014

Witt kernels and Brauer kernels for quartic extensions in characteristic two

Let $F$ be a field of characteristic $2$ and let $E/F$ be a field extension of degree $4$. We determine the kernel $W_q(E/F)$ of the restriction map $W_qF\to W_qE$ between the Witt groups of nondegenerate quadratic forms over $F$ and over $E$, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduct the corresponding result for the Witt kernel $W(E/F)$ of the restriction map $WF\to WE$ between the Witt rings of nondegenerate symmetric bilinear forms over $F$ and over $E$ from earlier results by the first author. As application, we describe the $2$-torsion part of the Brauer kernel for such extensions.

math.AC↗

arXiv2014

Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

Let $F$ be a field of characteristic $2$ and let $K/F$ be a purely inseparable extension of exponent $1$. We show that the extension is excellent for quadratic forms. Using the excellence we recover and extend results by Aravire and Laghribi who computed generators for the kernel $W_q(K/F)$ of the natural restriction map $W_q(F)\to W_q(K)$ between the Witt groups of quadratic forms of $F$ and $K$, respectively, where $K/F$ is a finite multiquadratic extension of separability degree at most $2$.

math.AC↗

arXiv2010

Dimensions of anisotropic indefinite quadratic forms II --- The lost proofs

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of anisotropic forms over F that satisfy certain additional properties. We construct various examples of fields with infinite Hasse number and prescribed finite values of u that satisfy additional properties pertaining to the space of orderings of the field. We also construct to each positive integer n a real field F such such that the Hasse number is 2^{n+1} and such that each quadratic form over F of dimension 1+2^n is a Pfister neighbor. These results were announced (without proof) in the article "Dimensions of anisotropic indefinite quadratic forms II" by the present author.

math.RA↗