On $n$-universal quadratic forms over dyadic local fields

TL;DR

This paper establishes necessary and sufficient conditions for integral quadratic forms over dyadic local fields to be n-universal, introducing a minimal testing set analogous to classical universality theorems.

Contribution

It provides a complete characterization of n-universal quadratic forms over dyadic local fields using Beli's invariants, and proposes a minimal testing set similar to known universality theorems.

Findings

Characterization of n-universal forms over dyadic local fields

Development of a minimal testing set for n-universality

Extension of classical universality theorems to dyadic local fields

Abstract

Let $ n \ge 2$ be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be $ n $-universal by using invariants from Beli's theory of bases of norm generators. Also, we provide a minimal set for testing $ n $-universal quadratic forms over dyadic local fields, as an analogue of Bhargava and Hanke's 290-theorem (or Conway and Schneeberger's 15-theorem) on universal quadratic forms with integer coefficients.