Decomposable quadratic forms and involutions

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...

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Metabolic Involutions and Quadratic Radical Extensions

In this paper we characterise involutions that become metabolic over a quadratic field extension attained by adjoining a square root.

Quadratic Functions, Optimization, and Quadratic Forms