Totally decomposable quadratic pairs

Structure and Applications of Totally Decomposable Metrics

Decomposable Quadratic Forms and Involutions

In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic forms when its adjoint involution decomposes as a tensor product of involutions on central simple algebras. We give a positive answer for quadratic forms defined over local or global fields and produce counterexamples over fields of rational fractions in two variable...

Decomposable quadratic forms in Banach spaces

A continuous quadratic form on a real Banach space X is called decomposable if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if X belongs to a certain class of superreflexive Banach spaces, including all Lp(μ) spaces with 2 ≤ p < ∞, then each continuous quadratic form on X is decomposable. On the other hand, on each infinite-di...

On Totally Decomposable Algebras with Involution in Characteristic Two

A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogon...

Singular and Totally Singular Generalised Quadratic Forms

In this paper we present a decomposition theorem for generalised quadratic forms over a division algebra with involution in characteristic 2. This is a generalisation of a decomposition result on quadratic forms in characteristic 2 from [3] and extends a generalisation of the Witt decomposition theorem for nonsingular forms to cover forms that may be singular.