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Teodor Banica; Ion Nechita; Jean-Marc Schlenker
Analytic aspects of the circulant Hadamard conjecture
(Aspects analytiques de la conjecture d’Hadamard circulante)
Annales mathématiques Blaise Pascal, 21 no. 1 (2014), p. 25-59, doi: 10.5802/ambp.334
Article PDF | Reviews MR 3248220 | Zbl 1297.05042
Class. Math.: 05B20
Keywords: Circulant Hadamard matrix

Résumé - 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 $|q_0|=\ldots =|q_{N-1}|=1$ the quantity $\Phi =\sum _{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi \ge N^2$, 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.

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