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 language||English (US)|
|Journal||Journal of Integer Sequences|
|State||Published - Jun 18 2007|