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The Bass and Topological Stable Ranks of $H^\infty_\R(\D)$ and $A_\R(\D)$

In this note we prove that the Bass stable rank of $H^\infty_\R(\D)$ is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of $H^\infty_\R(\D)$ is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of $A_\R(\D)$ is two and the topological stable rank of $A_\R(\D)$ is two, settling a conjecture by the second author. We also present a $\bar\partial$-free proof of the second author's characterization of the reducible pairs in $A_\R(\D)$.