On 5/29/07, Michael Kleber <michael.kleber@gmail.com> wrote:
The "extended version", i.e. the one which actually contains proofs, is at http://www.math.unizh.ch/user/gmaze/Articles/detcirc2.pdf This paper explores the question of whether a circulant matrix whose entries are all {0,1} or all {-1,1} can have as large a determinant as is possible if you remove the "circulant" criterion (a.k.a. the Hadamard bound). I don't think it answers Fred's determinant question, but I just skimmed it.
Thanks to everyone who pitched in to help sort out my attack of dodgy viewer software! This question turns out to have been well investigated: in particular, a "circulant" (what I called cyclic) matrix of order n is singular just when the coefficients along a row, mapped onto a polynomial in the natural fashion, correspond to a multiple of a (cyclotomic) factor of x^n - 1. E.g. since x+1 divides x^4-1, this "binary" (0-1) matrix is singular over the reals: [0 1 1 0] [1 1 0 0] [1 0 0 1] [0 0 1 1] Fred Lunnon