Gauss congruences for rational functions in several variables

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.