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

Gary Greaves; Huu An Phan

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arXiv · 2511.08333

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

Gary Greaves·Huu An Phan

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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.

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BibTeX RIS

Gary Greaves, Huu An Phan. 2025-11-11. Characteristic polynomials of $\{\pm 1\}$-matrices modulo a power of $2$. https://arxiv.org/abs/2511.08333

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