Gauss congruences for rational functions in several variables

We investigate necessary as well as sufficient conditions under which the Laurent series coefficients fn associated to a multivariate rational function satisfy the Gauss congruences, that is, fmpr≡fmpr−1 (mod pr). 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 when Q is linear in each variable.