Rosenbrock's Theorem characterizes Pr\"{u}fer domains

Abstract

Under coprimality assumptions on certain submatrices, Rosenbrock's Theorem relates the Smith form of a matrix $P$ over an elementary divisor domain $\mathcal{R}$ to the Smith-McMillan form of a matrix $G$ over the field of fractions of $\mathcal{R}$, where $G$ is a Schur complement in $P$. If $\FR$ is not an elementary divisor domain, Rosenbrock's Theorem is not directly applicable in its original form, because not every matrix is unimodularly equivalent to a matrix in Smith form. In this paper, we state an ideal-theoretic version of Rosenbrock's Theorem that is meaningful over any integral domain, and we show that it is equivalent to the classic formulation over an elementary divisor domain. Moreover, we give a characterization of Pr\"{u}fer domains as those integral domains over which the ideal-theoretic version of Rosenbrock's Theorem holds for every matrix satisfying the assumptions. In particular, the theorem does not hold for every admissible matrix over $\mathbb{C}[x_1,\dots,x_d]$ when $d \geq 2$, but we show that it holds Zariski-generically when $d \leq 3$. Finally, we prove that, if $\mathcal{R}$ is an integral domain such that every right invertible matrix can be completed to a unimodular matrix, then every matrix $P$ that satisfies the assumptions of the ideal-theoretic Rosenbrock's Theorem and realizes the same Schur complement $G$ shares the same ideal-theoretic generalization of the Smith form.