Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker
2014 Annales mathématiques Blaise Pascal  
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 qiq k qj 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,
more » ... tical points of Φ, and (3) the computation of the moments of Φ. We explore here these questions, with some results and conjectures.
doi:10.5802/ambp.334 fatcat:l27s5h7ge5azlkpuwiysviwgbi