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Trees of definable sets over the p-adics
-
Immanuel Halupczok
Published/Copyright:
February 9, 2010
Abstract
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.
Received: 2008-06-27
Revised: 2009-02-02
Published Online: 2010-02-09
Published in Print: 2010-May
© Walter de Gruyter Berlin · New York 2010
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Articles in the same Issue
- On the Rosenberg-Zelinsky sequence in abelian monoidal categories
- Arakelov theory of noncommutative arithmetic surfaces
- The value distribution of additive arithmetic functions on a line
- Kähler-Ricci solitons on homogeneous toric bundles
- The Jiang–Su algebra revisited
- Trees of definable sets over the p-adics
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Articles in the same Issue
- On the Rosenberg-Zelinsky sequence in abelian monoidal categories
- Arakelov theory of noncommutative arithmetic surfaces
- The value distribution of additive arithmetic functions on a line
- Kähler-Ricci solitons on homogeneous toric bundles
- The Jiang–Su algebra revisited
- Trees of definable sets over the p-adics
- A new series of compact minitwistor spaces and Moishezon twistor spaces over them
