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A Mahler's theorem for functions from words to integers

Authors Jean-Eric Pin, Pedro V. Silva

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Jean-Eric Pin
Pedro V. Silva

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Jean-Eric Pin and Pedro V. Silva. A Mahler's theorem for functions from words to integers. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 585-596, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


In this paper, we prove an extension of Mahler's theorem, a celebrated result of $p$-adic analysis. Mahler's original result states that a function from $N$ to $Z$ is uniformly continuous for the $p$-adic metric $d_p$ if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from $A^*$ to $Z$, where $d_p$ is now the profinite metric defined by $p$-groups (pro-$p$ metric).
  • $p$-adic topology
  • binomial coefficients
  • Mahler's theorem
  • $p$-group languages


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