  Let $K$ be a field, complete with respect to a discrete non-archimedian
valuation and let $k$ be the residue field. Consider a system $F$ of $n$
polynomial equations in $K\vars$. Our first result is a reformulation of the
classical Hensel's Lemma in the language of tropical geometry: we show
sufficient conditions (semiregularity at $w$) that guarantee that the first
digit map $\delta:(K^\ast)^n\to(k^\ast)^n$ is a one to one correspondence
between the solutions of $F$ in $(K^\ast)^n$ with valuation $w$ and the
solutions in $(k^\ast)^n$ of the initial form system ${\rm in}_w(F)$. Using
this result, we provide an explicit formula for the number of solutions in
$(K^\ast)^n$ of a certain class of systems of polynomial equations (called
regular), characterized by having finite tropical prevariety, by having initial
forms consisting only of binomials, and by being semiregular at any point in
the tropical prevariety. Finally, as a consequence of the root counting
formula, we obtain the expected number of roots in $(K^\ast)$ of univariate
polynomials with given support and random coefficients.
