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

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 $49$.