Trees of definable sets over the p-adics

Immanuel Halupczok 1
  • 1 Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany. e-mail:


To a definable subset of (or to a scheme of finite type over ℤp) one can associate a tree in a natural way. It is known that the corresponding Poincaré series ∑NλZλ ∈ ℤ[[Z]] is rational, where Nλ is the number of nodes of the tree at depth λ. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of , and for one-dimensional sets.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.