  We investigate necessary as well as sufficient conditions under which the
Laurent series coefficients $f_{\boldsymbol{n}}$ associated to a multivariate
rational function satisfy Gauss congruences, that is $f_{\boldsymbol{m}p^r}
\equiv f_{\boldsymbol{m}p^{r-1}}$ modulo $p^r$. For instance, we show that
these congruences hold for certain determinants of logarithmic derivatives. As
an application, we completely classify rational functions $P/Q$ satisfying the
Gauss congruences in the case that $Q$ is linear in each variable.
