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arXiv · 2105.14057

An Isometric Invariant of Quadratic Spaces over Finite Fields

Semin Yoo
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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 $(\mathbb{F}_{q}^{n},\text{dot}_{n})$.

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Semin Yoo. 2021-12-27. An Isometric Invariant of Quadratic Spaces over Finite Fields. https://arxiv.org/abs/2105.14057

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