Related papers: Algorithms for quadratic forms over global functio…
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…
It is well known that every non-degenerate quadratic form admits a decomposition into an orthogonal sum of its anisotropic part and a hyperbolic form. This decomposition is unique up to isometry. In this paper we present an algorithm for…
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 extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in…
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…
We study anisotropic universal quadratic forms over semi-global fields; i.e., over one-variable function fields over complete discretely valued fields. In particular, given a semi-global field $F$, we compute both the $m$-invariant of $F$…
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.
We propose a randomized polynomial time algorithm for computing nontrivial zeros of quadratic forms in 4 or more variables over $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is a finite field of odd characteristic. The algorithm is based on a…
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…
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,…
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 aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given…
The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in…
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 algorithms for computing the length of a sum of squares and a Pythagoras element in a global field $K$ of characteristic different from $2$. In the first part of the paper, we present algorithms for computing the length…
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued…
We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can…
Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
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…