Gauss congruences for rational functions in several variables

TL;DR

This paper studies conditions under which coefficients of multivariate rational functions satisfy Gauss congruences, providing classifications and identifying cases where these congruences hold, especially for functions with linear denominators.

Contribution

It establishes necessary and sufficient conditions for Gauss congruences in multivariate rational functions and classifies functions satisfying these congruences when the denominator is linear.

Findings

Gauss congruences hold for certain determinants of logarithmic derivatives.

Complete classification of rational functions with linear denominators satisfying Gauss congruences.

Identification of conditions ensuring coefficients follow Gauss congruences.

Abstract

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.