Characteristic polynomials of $\{\pm 1\}$-matrices modulo a power of $2$

TL;DR

This paper determines the number of congruence classes of characteristic polynomials of large symmetric and skew-symmetric -matrices modulo powers of 2, solving a conjecture and introducing new graph concepts.

Contribution

It provides exact counts of characteristic polynomial classes modulo 2^e for symmetric and skew-symmetric -matrices, and introduces the concepts of lift graphs and walk polynomials.

Findings

Number of classes for symmetric matrices: 2^{inom{e-2}{2}} or 2^{inom{e-2}{2}+1}

Number of classes for skew-symmetric matrices: 2^{loor{rac{e-1}{2}}loor{rac{e-2}{2}}} or 2^{loor{rac{e-2}{2}}loor{rac{e-3}{2}}}

Introduction of lift graphs and walk polynomials as main tools.

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.