Hyperbolicity over function fields of quadrics

TL;DR

This paper explores the conditions under which quadratic forms become hyperbolic over function fields, providing new characterizations and simplified proofs of key results in quadratic form theory.

Contribution

It introduces a new characterization of hyperbolicity over function fields and offers novel, concise proofs of established quadratic form results.

Findings

Quadratic forms become hyperbolic over their function fields under certain conditions.

New, elementary proofs of classical results in quadratic form theory.

A characterization of hyperbolicity over function fields is established.

Abstract

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 important established results in quadratic form theory by means of novel, short and elementary proofs.