Analytic aspects of the circulant Hadamard conjecture
Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 25-59.

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 Φ= i+k=j+l q i q k q j q l satisfies ΦN 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 Φ, (2) the study of the critical points of Φ, and (3) the computation of the moments of Φ. We explore here these questions, with some results and conjectures.

On étudie la question de comptage pour les matrices d’Hadamard réelles ou complexes circulantes, en utilisant des méthodes analytiques. Notre remarque principale est que pour |q 0 |=...=|q N-1 |=1 la quantité Φ= i+k=j+l q i q k q j q l satisfait ΦN 2 , avec égalité si et seulement si q=(q i ) est le vecteur des valeurs propres d’une matrice d’Hadamard complexe circulante. Ceci suggère trois problèmes analytiques, à savoir : (1) la minimisation directe de Φ, (2) l’étude des points critiques de Φ, et (3) le calcul des moments de Φ. On explore ici ces questions, avec plusieurs résultats et conjectures.

DOI: 10.5802/ambp.334
Classification: 05B20
Keywords: Circulant Hadamard matrix
Teodor Banica 1; Ion Nechita 2; Jean-Marc Schlenker 3

1 Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France
2 CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France
3 University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg
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Teodor Banica; Ion Nechita; Jean-Marc Schlenker. Analytic aspects of the circulant Hadamard conjecture. Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 25-59. doi : 10.5802/ambp.334. https://ambp.centre-mersenne.org/articles/10.5802/ambp.334/

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