Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices

Abstract

Let H be a Hadamard Circulant matrix of order n=4h2 where h>1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial Φn(t) are bounded by a constant independent of n. Then for all the φ(n)n-th primitive roots w of 1, P(w)n is not an algebraic integer in the cyclotomic field K=Q(w), where P(t) is the representer polynomial of H and φ is the Euler function. This implies that P(w) is not a real number.