A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let $b_{1},\dots ,b_{g},x$ be regular nonunits in a local (Noetherian) ring $(R,M)$ and assume that $I$ $\subseteq$ $(xR)_{a}$, the integral closure of $xR$, where $I$ $=$ $(b_{1},\dots ,b_{g},x)R$. Then the main result shows that for all but finitely many units $u_{1},\dots ,u_{g}$ in $R$ that are non-congruent modulo $M$ and for all large integers $n$ and $k$ it holds that $I^{jn}$ $=$ $I^{[j]n}$ for $j$ $=$ $1,\dots ,k$ and $j$ not divisible by $char(R/M)$, where $I^{[j]}$ is the $j$-th bracket power $((b_{1}+u_{1}x)^{j}, \dots ,(b_{g}+u_{g}x)^{j},x^{j})R$ of $I$ $=$ $(b_{1}+u_{1}x, \dots ,b_{g}+u_{g}x,x)R$. And, conversely, if there exist positive integers $g$, $n$, and $k$ $\ge$ ${\binom{{n+g} }{{g}}}$ such that $I$ has a basis $\beta _{1},\dots ,\beta _{g} ,x$ such that $I^{kn}$ $=$ $({\beta _{1}}^{k},\dots ,{\beta _{g}}^{k},x^{k})^{n}R$, then $I$ has analytic spread one.