An Isometric Invariant of Quadratic Spaces over Finite Fields

TL;DR

This paper introduces a new isometric invariant for quadratic spaces over finite fields, providing insights into subspace embeddings and offering a novel proof of Minkowski's sphere size formula.

Contribution

It constructs a new invariant for quadratic spaces over finite fields and applies it to analyze subspace embeddings and reprove Minkowski's formula.

Findings

New invariant characterizes quadratic space types

Provides criteria for embedding quadratic subspaces

Offers a new proof of Minkowski's sphere size formula

Abstract

Let $\mathbb{F}_{q}$ be the finite field with an odd prime power $q$. In this paper, we construct a new isometric invariant of combinatorial type on $(\mathbb{F}^{n}_{q},\text{dot}_{n})$, where $\text{dot}_{n}(\mathbf{x}):=x_{1}^{2}+\cdots+x_{n}^{2}$. Additionally, using counts from our new invariant, we give a new proof of Minkowski's formula on the size of spheres over finite fields. We also show which types of quadratic subspaces can be embedded in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$.