Analytic aspects of the circulant Hadamard conjecture

TL;DR

This paper explores analytic methods to count circulant Hadamard matrices, focusing on a key quantity related to eigenvalue vectors, and proposes three mathematical problems to advance understanding.

Contribution

It introduces an analytic framework for studying circulant Hadamard matrices, including the analysis of a specific sum and posing related optimization and moment problems.

Findings

The quantity  satisfies  bgeq N^2 with equality for eigenvalue vectors of rescaled circulant Hadamard matrices.

Proposes three analytic problems: minimization of , critical point analysis, and moment computation.

Provides initial results and conjectures on these analytic problems.

Abstract

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here these questions, with some results and conjectures.