On positive Matrices which have a Positive Smith Normal Form

TL;DR

This paper generalizes a known result about positive semi-definite symmetric matrices over real polynomial rings to matrices over more general algebraic domains, establishing conditions for their Smith normal form to be positive.

Contribution

It extends the positivity property of Smith normal forms from polynomial rings to symmetric matrices over formally real principal domains, under specific positivity and prime-related hypotheses.

Findings

Smith normal form has positive diagonal coefficients under certain conditions.

Counterexamples show the necessity of the additional hypothesis.

Partial extension to Dedekind domains provided.

Abstract

It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, 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.