Simultaneous Stabilization in $A_\mathbb{R}(\mathbb{D})$

In this note we study the problem of simultaneous stabilization for the algebra $A_\R(\D)$. Invertible pairs $(f_j,g_j)$, $j=1,..., n$, in a commutative unital algebra are called \textit{simultaneously stabilizable} if there exists a pair $(α,β)$ of elements such that $αf_j+βg_j$ is invertible in this algebra for $j=1,..., n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\R(\D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\R(\D)$. For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\R(\D)^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_\R(\D)$ such that $uf+vg=1$.

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