Wild partitions and number theory

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Abstract

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λQ,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function ΛQ,(x) = ∑λQ,nx n and use it to show that λQ,n1/n tends to Q1/(p-1). We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture contrasts sharply with known results for function fields, and our arguments explain this contrast.

Original languageEnglish (US)
Article number07.6.6
JournalJournal of Integer Sequences
Volume10
Issue number6
StatePublished - Jun 18 2007

Keywords

  • Mass
  • Partition
  • Ramified
  • Wild
  • p-adic

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