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Some determinants involving binary forms

Yue-Feng She, Zhi-Wei Sun

TL;DR

The paper investigates determinants built from binary quadratic forms via Jacobi symbols, proving that for odd $n>1$ with $(d/n)=-1$, one has $\varphi(n)^2\mid [c,d]_n$, thus resolving a conjecture of Sun. The authors develop several determinant identities using the Weinstein–Aronszajn identity and the Matrix-Determinant Lemma, complemented by Möbius inversion and the Chinese Remainder Theorem, to handle Jacobi-symbol evaluations. They derive a compact modulo-$p$ formula for determinants of the form $\det[P(ij^{-1})]$ depending on coefficients of $P$, and apply it to evaluate $D_p^-(1,1)$ in several residue classes modulo $p$, obtaining precise congruences and Legendre-symbol relations. The results connect determinants of binary quadratic forms with arithmetic over finite fields, providing explicit criteria across multiple modular regimes and supporting related conjectures and patterns.

Abstract

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.

Some determinants involving binary forms

TL;DR

The paper investigates determinants built from binary quadratic forms via Jacobi symbols, proving that for odd with , one has , thus resolving a conjecture of Sun. The authors develop several determinant identities using the Weinstein–Aronszajn identity and the Matrix-Determinant Lemma, complemented by Möbius inversion and the Chinese Remainder Theorem, to handle Jacobi-symbol evaluations. They derive a compact modulo- formula for determinants of the form depending on coefficients of , and apply it to evaluate in several residue classes modulo , obtaining precise congruences and Legendre-symbol relations. The results connect determinants of binary quadratic forms with arithmetic over finite fields, providing explicit criteria across multiple modular regimes and supporting related conjectures and patterns.

Abstract

In this paper, we study arithmetic properties of certain determinants involving powers of , where and are integers. For example, for any odd integer with we prove that is divisible by , where is the Jacobi symbol and is Euler's totient function. This confirms a previous conjecture of the second author.
Paper Structure (5 sections, 8 theorems, 72 equations)

This paper contains 5 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['cdp']}
  3. Proof of Theorem \ref{['shift']}
  4. Proof of Theorem \ref{['formula']}
  5. Deduce Corollary \ref{['Dp^-(1,1)']} from Theorem \ref{['formula']}

Key Result

Theorem 1.1

Let $c,d\in\mathbb Z$. For any odd number $n>1$ with $(\frac{d}{n})=-1$, we have $\varphi(n)^2\mid [c,d]_n$, where $\varphi$ is Euler's totient function.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Corollary 1.2
  • Conjecture 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 4.1: The Matrix-Determinant Lemma