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Let $\frak{a}$ be a proper ideal of a commutative Noetherian ring
$R$ of prime characteristic $p$ and let $Q(a)$ be the smallest
positive integer $m$ such that
${(\frak{a}^{F})}^{[p^m]}=\frak{a}^{[p^m]}$, where $\frak{a}^F$ is
the Frobenius closure of $\frak{a}$. This paper is concerned with
the question whether the set $\{Q(\frak{a} ^{[p^m]}): m \in
\mathbb{N}_0\}$ is bounded. We give an affirmative answer in the
case that the ideal $n$ is generated by an u.s.d-sequence $c_{1},
... , c_{n}$ for $R$ such that
\noindent (i) the map $R/ \Sigma^{n}_{j=1} R{c}_{j}\rightarrow R/
\Sigma^{n}_{j=1} R{c}_{j}^{2}$ induced by multiplication by
$c_{1},
... , c_{n}$ is an $R$-monomorphism;\(ii) for all $\frak{p} \in$ ass$(c_{1}^{j}, ... , c_{n}^{j}),
c_{1}/1,...,c_{n}/1$ is a $\frak{p}R_\frak{p}$-filter regular
sequence for $R_\frak{p}$ for $j \in \{1, 2\}$.
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