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

In this note we study the problem of simultaneous stabilization for the algebra $A_{\mathbb{R}}(\text{D})$. Invertible pairs $(f_j,g_j)$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},n$, in a commutative unital algebra are called $\mathit{\text{simultaneously}}\mathit{\text{stabilizable}}$ if there exists a pair $(\alpha ,\beta )$ of elements such that $\alpha f_j+\beta g_j$isinvertibleinthisalgebrafor$j=1,…, n$. 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 $A_{\mathbb{R}}(\text{D})$ 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 $A_{\mathbb{R}}(\text{D})$. For $n\ge 3$ 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 $A_{\mathbb{R}}(\text{D}{)}^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_{\mathbb{R}}(\text{D})$ such that $uf+vg=1$.