Related papers: Isotropy of quadratic forms over function fields i…

We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric.

Let $\varphi$ and $\psi$ be quadratic forms over a field $K$ of characteristic different from 2. In this paper, we give a criterion for isotropy of $\varphi$ over the function field of $\psi$ in terms of representations and we apply it to…

A strong consequence of quadratic forms becoming hyperbolic over the function field of a form is established. This result is invoked to obtain a new characterisation of hyperbolicity over function fields, and to recover a number of…

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in arXiv:1304.0708 to global function fields of odd characteristics. First, we present algorithm for checking if a given…

In this note, first we show that there is no stable quadratic polynomial over finite fields of characteristic two and then show that there exist stable quadratic polynomials over function fields of characteristic two.

We present a complete suite of algorithms for finding isotropic vectors of quadratic forms (of any dimension) over an arbitrary global field of characteristic different from 2. This is a new version with numerous changes and improvements.

We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and…

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of…

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…

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…

We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation…

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…

For quadratic forms in $4$ variables defined over the rational function field in one variable over $\mathbb C(\!(t)\!)$, the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is…

We investigate diagonal forms of degree $d$ over the function field $F$ of a smooth projective $p$-adic curve: if a form is isotropic over the completion of $F$ with respect to each discrete valuation of $F$, then it is isotropic over…

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…

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary…

This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.

The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted…