Analytic aspects of the circulant Hadamard conjecture

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 (2014)

  • Volume: 21, Issue: 1, page 25-59
  • ISSN: 1259-1734

Abstract

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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 Φ ≥ 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.

How to cite

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Banica, Teodor, Nechita, Ion, and Schlenker, Jean-Marc. "Analytic aspects of the circulant Hadamard conjecture." Annales mathématiques Blaise Pascal 21.1 (2014): 25-59. <http://eudml.org/doc/275493>.

@article{Banica2014,
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.},
affiliation = {Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France; CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France; University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg},
author = {Banica, Teodor, Nechita, Ion, Schlenker, Jean-Marc},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Circulant Hadamard matrix; Hadamard matrix; circulant Hadamard matrix; complex Hadamard matrix},
language = {eng},
month = {1},
number = {1},
pages = {25-59},
publisher = {Annales mathématiques Blaise Pascal},
title = {Analytic aspects of the circulant Hadamard conjecture},
url = {http://eudml.org/doc/275493},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Banica, Teodor
AU - Nechita, Ion
AU - Schlenker, Jean-Marc
TI - Analytic aspects of the circulant Hadamard conjecture
JO - Annales mathématiques Blaise Pascal
DA - 2014/1//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 1
SP - 25
EP - 59
AB - 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.
LA - eng
KW - Circulant Hadamard matrix; Hadamard matrix; circulant Hadamard matrix; complex Hadamard matrix
UR - http://eudml.org/doc/275493
ER -

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