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Gauss congruences for rational functions in several variables

Volume 184 / 2018

Frits Beukers, Marc Houben, Armin Straub Acta Arithmetica 184 (2018), 341-362 MSC: 11A07, 11B37. DOI: 10.4064/aa170614-13-7 Published online: 20 August 2018

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 the Gauss congruences, that is, $f_{\boldsymbol{m}p^r} \equiv f_{\boldsymbol{m}p^{r - 1}} ({\rm mod}\ {p^r})$. 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 when $Q$ is linear in each variable.

Authors

  • Frits BeukersDepartment of Mathematics
    Utrecht University
    P.O. Box 80.010
    3508 TA Utrecht, The Netherlands
    e-mail
  • Marc HoubenDepartment of Mathematics
    Utrecht Universit
    P.O. Box 80.010
    3508 TA Utrecht, The Netherlands
    e-mail
  • Armin StraubDepartment of Mathematics and Statistics
    University of South Alabama
    411 University Blvd N
    Mobile, AL 36688, U.S.A.
    e-mail

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