The Pell sequence and cyclotomic matrices involving squares over finite fields

TL;DR

This paper explores properties of cyclotomic matrices over finite fields using Pell sequences and p-adic tools, revealing conditions for matrix singularity and determinants related to Pell sequence terms.

Contribution

It introduces new connections between Pell sequences, cyclotomic matrices, and finite field properties, providing criteria for matrix singularity and determinant conditions.

Findings

The matrix B_q((q-3)/2) is singular for q=p^f with f≥2.

Determinant of B_p((p-3)/2) is zero iff Q_p ≡ 2 mod p^2.

Established links between Pell sequence terms and matrix properties over finite fields.

Abstract

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.