Trees of definable sets over the p-adics

TL;DR

This paper explores the structure of trees associated with definable sets over p-adic integers, proposing a combinatorial classification and verifying it in specific cases, linking tree properties to rational Poincare series.

Contribution

It introduces a conjectural combinatorial description of trees from definable p-adic sets and confirms it in certain cases, advancing understanding of their structure.

Findings

The Poincare series of these trees is rational.

Any tree in the proposed class can be realized by a definable set.

The classification holds under weak smoothness, for Z_p^2, and one-dimensional sets.

Abstract

To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. 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 Z_p^2, and for one-dimensional sets.