Four-dimensional quadratic forms over $\mathbb C(\!(t)\!)(X)$

TL;DR

This paper investigates the validity of the local-global principle for isotropy of four-variable quadratic forms over the field (t) ext{(X)}, analyzing how different sets of valuations affect this principle.

Contribution

It provides new insights into the local-global principle for quadratic forms over complex function fields with respect to various valuation sets.

Findings

The local-global principle holds under certain valuation conditions.

Counterexamples are identified for specific valuation sets.

The results extend understanding of quadratic forms over complex function fields.

Abstract

For quadratic forms in $4$ variables defined over the rational function field in one variable over $\mathbb C(\!(t)\!)$, the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is examined.