Related papers: On the Pfister Number of Quadratic Forms

For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on…

Let $q$ be a unimodular quadratic form over a field $K$. Pfister's famous local--global principle asserts that $q$ represents a torsion class in the Witt group of $K$ if and only if it has signature $0$, and that in this case, the order of…

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, also verifying a…

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class…

Let $F$ be a field of characteristic $2$, $\pi$ be an $n$-fold bilinear Pfister form over $F$ and $\varphi$ an arbitrary quadratic form over $F$. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy…

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…

We study linkage of $(d+1)$-fold quadratic Pfister forms over function fields in one variable over a henselian valued field of 2-cohomological dimension $d$. Specifically, we characterise this property in terms of linkage of quadratic…

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…

The isotropy of multiples of Pfister forms is studied. In particular, an improved lower bound on the value of their first Witt index is obtained. This result and certain of its corollaries are applied to the study of the weak isotropy index…

We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

An alternative construction, using Witt's formalism, of the Arf-invariant of quadratic forms in characteristic 2.

We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of…

We want to bound the symbol length of classes in ${_{2^{m-1}}Br}(F)$ which are represented by tensor products of 5 or 6 cyclic algebras of degree $2^m$. The main ingredients are the chain lemma for quadratic forms, a form of a generalized…

We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then…

Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer…

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…

In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for…