Eigenvalues of Circulant Matrices and a Conjecture of Ryser

Reinhardt Euler, Luis H. Gallardo, Olivier Rahavandrainy

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under some linear conditions on the $h_{i}$'s. All these conditions hold in the known case $n=4,$ so that our results can be thought as characterizations of properties that only hold when $n=4.$ Our first conditions imply that some eigenvalue $\lambda$ of $H$ is a sum of $\sqrt{n}$ terms $h_{j}\omega^{j}$, where $\omega$ is a primitive $n$-th root of $1$. The same conclusion holds also if some complex arithmetic means associated to $\lambda$ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of \emph{robust} Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very difficult to be able to prove the result.