  We prove the multivariate Fujiwara bound for exponential sums: for a
$d$-variate exponential sum $f$ with scaling parameter $\mu$, if $x$ is
contained in the amoeba $\mathscr{A}(f)$, then the distance from $x$ to the
Archimedean tropical variety associated to $f$ is at most $d \sqrt{d}\, 2\log(2
+ \sqrt{3})/ \mu$. If $f$ is polynomial, then the bound can be improved to $d
\log(2 + \sqrt{3})$.
