On Positive Matrices Which Have a Positive Smith Normal Form

It is known that any symmetric matrix M with entries in R[x] and which is positive semi-definite for any substitution of x ∈ R, has a Smith normal form whose diagonal coefficients are constant sign polynomials in R[x]. We generalize this result by considering a symmetric matrix M with entries in a formally real principal domain A, we assume that M is positive semi-definite for any ordering on A and, under one additionnal hypothesis concerning non-real primes, we show that the Smith normal of M is positive, up to association. Counterexamples are given when this last hypothesis is not satisfied. We give also a partial extension of our results to the case of Dedekind domains.