$p$-adic Zeros of Systems of Quadratic Forms

TL;DR

This survey reviews research on the minimum number of variables needed for systems of quadratic forms over p-adic fields to have a non-trivial common zero.

Contribution

It compiles and discusses existing results on variable bounds for quadratic forms over p-adic fields, highlighting key developments and open problems.

Findings

Established bounds for the number of variables needed.

Identified gaps in current understanding.

Summarized recent advances in the field.

Abstract

This survey describes work on the number of variables required to ensure that a system of r quadratic forms over the p-adics has a non-trivial common zero.