A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

Abstract: Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}{F}(f;x{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|{F}^{-s}$ is integrable near $x{0}\in X$. We show that $\mathrm{lct}{F}(f;x{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}{F}(f;x{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$.