Related papers: Four-dimensional quadratic forms over $\mathbb C(\…

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 prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of transcendence degree at least 2 over algebraically closed fields. Our construction involves…

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $\Omega_R$ be the set of rank 1 discrete valuations of $L$…

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…

We use the Witt index to define and study a refined notion of the local-global principle for isotropy of quadratic forms over a field $k$ and to define and study refined versions of the $m$-invariant of $k$. We also explore connections…

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…

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…

We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on…

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 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…

We investigate the Hasse principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Over purely transcendental field extensions of fields that…

We investigate the probability that a random quadratic form in ${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result…

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…

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 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.

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using…

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the…

We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the…

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

We show that the density of quadratic forms in $n$ variables over ${\mathbb Z}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a…