Wild Partitions and Number Theory
Wild Partitions and Number Theory
Summary: 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 \(\lambda_{Q,n}\) counting the number of certain wild partitions of \(n\) . We give an explicit formula for the corresponding generating function \(\Lambda_Q(x) = \sum \lambda_{Q,n} x^n\) and use it to show that \(\lambda^{1/n}_{Q,n}\) tends to \(Q^{1/(p-1)}\). 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.
- University of Minnesota-Twin Cities United States
Ramification and extension theory, Combinatorial aspects of partitions of integers, Ramified, Number Theory, Wild, Elementary theory of partitions, Mass, p-adic, Other number fields, Partition, 510
Ramification and extension theory, Combinatorial aspects of partitions of integers, Ramified, Number Theory, Wild, Elementary theory of partitions, Mass, p-adic, Other number fields, Partition, 510
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