Characteristic polynomials of $\{\pm 1\}$-matrices modulo a power of $2$
Gary Greaves, Huu An Phan
TL;DR
This work determines the exact number of congruence classes modulo 2^e of characteristic polynomials for three natural families of ±1 matrices with constant diagonal: symmetric S_n, skew-symmetric T_n, and general unit-up U_n, for all sufficiently large n. The authors introduce lift graphs and lift tournaments as the key constructive devices, paired with the walk polynomial to certify the existence of appropriate lifts that realize all target residue patterns. They prove sharp counts: |C_e(U_n)| = 2^{inom{e}{2}} for large n, |C_e(S_n)| equals 2^{inom{e-2}{2}} or 2^{inom{e-2}{2}+1} depending on n's parity, and |C_e(T_n)| equals 2^{ig floor ( rac{e-1}{2})ig floorig floor imes ig floor ( rac{e-2}{2})ig floor} or 2^{ig floor ( rac{e-2}{2})ig floorig floor imes ig floor ( rac{e-3}{2})ig floor} with parity-based distinctions. The results resolve Greaves and Yatsyna’s conjecture and reveal a coherent 2-adic lifting pattern across symmetric, skew-symmetric, and unit-on-diagonal classes. Methodologically, the lift graph/tournament framework, together with the walk polynomial, provides a versatile and elementary approach that yields precise, constructive counts with recursive bounds on the minimal n required. The findings have implications for arithmetic properties of ±1 matrices and related combinatorial matrix theory, including connections to Seidel-type matrices and equiangular lines.
Abstract
For a fixed integer $e \geqslant 3$ and $n$ large enough, we show that the number of congruence classes modulo $2^e$ of characteristic polynomials of $n \times n$ symmetric $\{\pm 1\}$-matrices with constant diagonal is equal to $2^{\binom{e-2}{2}}$ if $n$ is even or $2^{\binom{e-2}{2}+1}$ if $n$ is odd, thereby solving a conjecture of Greaves and Yatsyna from 2019. We also show that, for $n$ large enough, the number of congruence classes modulo $2^e$ of characteristic polynomials of $n \times n$ skew-symmetric $\{\pm 1\}$-matrices with constant diagonal is equal to $2^{\lfloor \frac{e-1}{2} \rfloor\lfloor \frac{e-2}{2} \rfloor}$ if $n$ is even or $2^{\lfloor \frac{e-2}{2} \rfloor\lfloor \frac{e-3}{2} \rfloor}$ if $n$ is odd. We introduce the concept of a lift graph/tournament, which serves as our main tool. We also introduce the notion of the walk polynomial of a graph, which enables us to show the existence of the requisite lift tournaments.