[math-fun] Circulant determinants

A n x n "circulant" determinant A = |A_ij| is constructed from n-vector S = [S_j] by setting A_{i,j} = S_{(i+j) mod n}. Let S be a random n-vector with elements in {-1,+1}. What is the probability that A = 0 ? Example: when n = 32, A has a maximum about 23 decimal digits; yet about 30% of the time it turns out to be zero! Remark: A well-known classical theorem states that A = \product_i \sum_j S_j w^{i j} where w is a primitive n-th root of unity. Fred Lunnon