Characteristic polynomials of \± 1\-matrices modulo a power of 2

2025/11/11 by Greaves, Gary, Phan, Huu An
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.08333

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 × n symmetric \± 1\-matrices with constant diagonal is equal to 2^\binome-22 if n is even or 2^\binome-22+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 × n skew-symmetric \± 1\-matrices with constant diagonal is equal to 2^{\lfloor (e-1)/(2) \rfloor\lfloor (e-2)/(2) \rfloor} if n is even or 2^{\lfloor (e-2)/(2) \rfloor\lfloor (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.

Citations

• Determinants of Seidel Tournament Matrices
• Equiangular lines in Euclidean spaces: dimensions 17 and 18
• Equiangular lines in low dimensional Euclidean spaces

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