On positive Matrices which have a Positive Smith Normal Form

Abstract

It is known that any symmetric matrix $M$ with entries in $\mathbb{R}[x]$ and which is positive semi-definite for any substitution of $x\in \mathbb{R}$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\mathbb{R}[x]$. 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.