Square-reflexive polynomials

TL;DR

This paper characterizes when quadratic forms over rational function fields satisfy a local-global principle using polynomials, providing new elementary proofs for finite or pseudo-algebraically closed fields.

Contribution

It introduces a polynomial-based characterization for local-global principles of quadratic forms over certain function fields, simplifying existing proofs.

Findings

Characterization of quadratic forms satisfying local-global principles

Elementary proofs for finite or pseudo-algebraically closed fields

Examples illustrating the polynomial criteria

Abstract

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 quadratic forms over $E(X)$ satisfy a local-global principle with respect to discrete valuations that are trivial on $E$. In this way new elementary proofs for the local-global principle are achieved in the cases where $E$ is finite or pseudo-algebraically closed. The study is complemented by various examples.