A Determinant Congruence Conjectured by Sun

TL;DR

This paper proves a strengthened form of Sun's conjecture on the divisibility of a determinant associated with a binary quadratic form, involving prime and composite cases, and uses algebraic and number-theoretic techniques.

Contribution

It establishes a new, stronger divisibility result for a quadratic form determinant conjectured by Sun, including prime and composite cases, with novel polynomial and rank analysis methods.

Findings

Determinant divisible by n^2 for composite n with no conditions on c,d

For prime p, determinant divisibility depends on Legendre symbol of d

Rank estimates and involution symmetry explain divisibility properties

Abstract

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2} \] with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\leg{d}{p}$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p$. The required rank defect follows from a coefficient cancellation obtained from the involution $t\mapsto d/t$ on $\Fp^\times$ and the condition $\leg{d}{p}=-1$.