On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

TL;DR

This paper establishes modular restrictions on the characteristic polynomial of Seidel matrices and applies these results to improve the upper bound on equiangular lines in 17-dimensional space from 50 to 49.

Contribution

It introduces new modular constraints on Seidel matrix polynomials and uses them to refine bounds on equiangular line configurations.

Findings

Derived restrictions modulo powers of 2 on characteristic polynomial coefficients.

Showed at most 2^{binom{e-2}{2}} possibilities for even order Seidel matrices.

Improved the upper bound for equiangular lines in R^{17} from 50 to 49.

Abstract

For $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial $\chi_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) possibilities for the congruence class of $\chi_S(x)$ modulo $2^e\mathbb Z[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in $\mathbb R^{17}$, that is, we reduce the known upper bound from $50$ to $49$.