Minimal rank of primitively $n$-universal integral quadratic forms over local rings

TL;DR

This paper determines the minimal rank of primitively n-universal integral quadratic forms over local rings, extending previous results for the case n=1 to all positive n and various local rings.

Contribution

It completely characterizes the minimal rank of primitively n-universal quadratic forms over local rings for all positive n and specific ring conditions.

Findings

Minimal rank for primitively 1-universal forms over $\\mathbb{Z}_p$ is 2 for odd p, and 3 for p=2.

Extended the minimal rank determination to all positive n over local rings where 2 is a unit or prime.

Provided a comprehensive classification of primitively n-universal quadratic forms over local rings.

Abstract

Let $F$ be a local field and let $R$ be its ring of integers. For a positive integer $n$, an integral quadratic form defined over $R$ is called primitively $n$-universal if it primitively represents all quadratic forms of rank $n$. It was proved in arXiv:2005.11268 that the minimal rank of primitively $1$-universal quadratic forms over the $p$-adic integer ring $\mathbb{Z}_p$ is $2$ if $p$ is odd, and $3$ otherwise. In this article, we completely determine the minimal rank of primitively $n$-universal quadratic forms over $R$ for any positive integer $n$ and any local ring $R$ such that $2$ is a unit or a prime.