关于具有正Smith Normal Form的正矩阵
On positive Matrices which have a Positive Smith Normal Form
摘要：
已知任何在$\R[x]$中具有元素并且对于$x\in\R$的任何替换都是半正定的对称矩阵$M$具有其对角系数是$\R[x]$中的常数符号多项式的Smith标准形。我们推广了这一结果，考虑一个对称矩阵$M$的条目在一个正式的实主域$A$，我们假设$M$是半正定的$A$上的任何顺序，并在一个额外的假设下，关于非实素数，我们证明了史密斯正常的$M$是积极的，直到协会。反例时，这最后一个假设是不满足的。我们还将所得结果部分推广到戴德金整环的情形。
英文摘要：
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.
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