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# Gauss congruences for rational functions in several variables

Gauss congruences for rational functions in several variables
Frits Beukers, Marc Houben, Armin Straub — Acta Arithmetica — Volume 184, 2018, Pages 341-362

## 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.

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