Analytic aspects of the circulant Hadamard conjecture

Teodor Banica; Ion Nechita; Jean-Marc Schlenker

doi:10.5802/ambp.334

Analytic aspects of the circulant Hadamard conjecture
[Aspects analytiques de la conjecture d’Hadamard circulante]

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

Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 25-59.

Résumé
Abstract

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 $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfait $Φ \geq N^{2}$ , 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.

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 $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfies $Φ \geq N^{2}$ , 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 $Φ$ , (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.

MR Zbl

DOI : 10.5802/ambp.334

Classification : 05B20
Keywords: Circulant Hadamard matrix
Mot clés : Matrice d’Hadamard circulante

Affiliations des auteurs :

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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     title = {Analytic aspects of the circulant {Hadamard} conjecture},
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     publisher = {Annales math\'ematiques Blaise Pascal},
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Teodor Banica; Ion Nechita; Jean-Marc Schlenker. Analytic aspects of the circulant Hadamard conjecture. Annales mathématiques Blaise Pascal, Tome 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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