Sun-type determinant and permanent congruences

TL;DR

This paper proves several of Sun's conjectures on determinants and permanents over residue classes modulo a prime, using advanced algebraic and combinatorial techniques.

Contribution

It confirms multiple conjectures by Sun, strengthening determinant criteria and establishing new congruences for permanents and determinants over finite fields.

Findings

Proved Sun's conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11, 4.12.

Established root-quotient criterion for irreducible binary quadratic forms.

Derived divisibility and congruence results for determinants and permanents.

Abstract

Sun proposed a collection of congruence and quadratic-residue conjectures for determinants and permanents over residue classes modulo a prime. This article proves Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list. The determinant part is strengthened to a root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field; the criterion gives the stated result for the determinant associated with the remaining binary quadratic form. The Cauchy-kernel part gives both derangement congruences modulo the square of the prime and a polynomial fixed-point permanent congruence modulo the prime. The Cayley-transform part gives the signed fixed-point determinant congruences, the quadratic-residue assertion for the signed derangement determinant, and the full fixed-point permanent congruence modulo the square of the prime. The half-size quadratic Cayley determinant is treated by a local expansion at a simple zero eigenvalue, giving divisibility by the square of the prime and, in the stronger congruence class, by its cube. The proofs combine finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence and first nonzero determinant terms.