Setting Work in Q p ( p -adics) Two-sorted language: Q p with - PowerPoint PPT Presentation

Trees of definable sets in Z p I. Halupczok Ecole Normale Superieure rue d’Ulm, Paris Barcelona Modnet conference 2008 Trees of definable sets in Z p 1 / 13 I. Halupczok

Setting • Work in Q p ( p -adics) • Two-sorted language: • Q p with field language • value group Z with ordered group language • valuation v : Q p → Z ∪ {∞} x ⊂ Q n • For tuples ¯ p , set v (¯ x ) := min { v ( x i ) | 1 ≤ i ≤ n } x ∈ Q n • Ball around ¯ p of (valuative) radius λ ∈ Z : x ′ | v (¯ x ′ − ¯ x + p λ Z n B (¯ x , λ ) := { ¯ x ) ≥ λ } = ¯ p • Recall: Balls of fixed radius λ form a partition of Q n p Trees of definable sets in Z p 2 / 13 I. Halupczok

Setting • Work in Q p ( p -adics) • Two-sorted language: • Q p with field language • value group Z with ordered group language • valuation v : Q p → Z ∪ {∞} x ⊂ Q n • For tuples ¯ p , set v (¯ x ) := min { v ( x i ) | 1 ≤ i ≤ n } x ∈ Q n • Ball around ¯ p of (valuative) radius λ ∈ Z : x ′ | v (¯ x ′ − ¯ x + p λ Z n B (¯ x , λ ) := { ¯ x ) ≥ λ } = ¯ p • Recall: Balls of fixed radius λ form a partition of Q n p Trees of definable sets in Z p 2 / 13 I. Halupczok

Setting • Work in Q p ( p -adics) • Two-sorted language: • Q p with field language • value group Z with ordered group language • valuation v : Q p → Z ∪ {∞} x ⊂ Q n • For tuples ¯ p , set v (¯ x ) := min { v ( x i ) | 1 ≤ i ≤ n } x ∈ Q n • Ball around ¯ p of (valuative) radius λ ∈ Z : x ′ | v (¯ x ′ − ¯ x + p λ Z n B (¯ x , λ ) := { ¯ x ) ≥ λ } = ¯ p • Recall: Balls of fixed radius λ form a partition of Q n p Trees of definable sets in Z p 2 / 13 I. Halupczok

Setting • Work in Q p ( p -adics) • Two-sorted language: • Q p with field language • value group Z with ordered group language • valuation v : Q p → Z ∪ {∞} x ⊂ Q n • For tuples ¯ p , set v (¯ x ) := min { v ( x i ) | 1 ≤ i ≤ n } x ∈ Q n • Ball around ¯ p of (valuative) radius λ ∈ Z : x ′ | v (¯ x ′ − ¯ x + p λ Z n B (¯ x , λ ) := { ¯ x ) ≥ λ } = ¯ p • Recall: Balls of fixed radius λ form a partition of Q n p Trees of definable sets in Z p 2 / 13 I. Halupczok

Setting • Work in Q p ( p -adics) • Two-sorted language: • Q p with field language • value group Z with ordered group language • valuation v : Q p → Z ∪ {∞} x ⊂ Q n • For tuples ¯ p , set v (¯ x ) := min { v ( x i ) | 1 ≤ i ≤ n } x ∈ Q n • Ball around ¯ p of (valuative) radius λ ∈ Z : x ′ | v (¯ x ′ − ¯ x + p λ Z n B (¯ x , λ ) := { ¯ x ) ≥ λ } = ¯ p • Recall: Balls of fixed radius λ form a partition of Q n p Trees of definable sets in Z p 2 / 13 I. Halupczok

Trees of sets in Z n p • Subballs of Z n p form a tree • Suppose ∅ � = X ⊂ Z n p . This yields a sub-tree T(X) of those balls intersecting X : • Vertices: T( X ) = { B = B (¯ x , λ ) | ¯ x ∈ X , λ ≥ 0 } • Root of T( X ) is Z p • Tree structure given by inclusion • B ⊂ Z n p ball, B ∩ X � = ∅ � T B (X) := subtree of T( X ) above B (i.e. vertices are balls contained in B ) Trees of definable sets in Z p 3 / 13 I. Halupczok

Trees of sets in Z n p • Subballs of Z n p form a tree • Suppose ∅ � = X ⊂ Z n p . This yields a sub-tree T(X) of those balls intersecting X : • Vertices: T( X ) = { B = B (¯ x , λ ) | ¯ x ∈ X , λ ≥ 0 } • Root of T( X ) is Z p • Tree structure given by inclusion • B ⊂ Z n p ball, B ∩ X � = ∅ � T B (X) := subtree of T( X ) above B (i.e. vertices are balls contained in B ) Trees of definable sets in Z p 3 / 13 I. Halupczok

Trees of sets in Z n p • Subballs of Z n p form a tree • Suppose ∅ � = X ⊂ Z n p . This yields a sub-tree T(X) of those balls intersecting X : • Vertices: T( X ) = { B = B (¯ x , λ ) | ¯ x ∈ X , λ ≥ 0 } • Root of T( X ) is Z p • Tree structure given by inclusion • B ⊂ Z n p ball, B ∩ X � = ∅ � T B (X) := subtree of T( X ) above B (i.e. vertices are balls contained in B ) Trees of definable sets in Z p 3 / 13 I. Halupczok

Examples of trees • Examples: p � Every node of T( X ) has p n children • X = Z n • X finite � each x ∈ X corresponds to infinite path in T( X ). . . • Infinite path in a tree T is a sequence of vertices { v 0 , v 1 , v 2 , . . . } ⊂ T with v 0 = root and v i +1 is child of v i • We have bijection 1:1 → ¯ { Infinite paths in T( X ) } ← X (= p -adic closure of X ) Trees of definable sets in Z p 4 / 13 I. Halupczok

Examples of trees • Examples: p � Every node of T( X ) has p n children • X = Z n • X finite � each x ∈ X corresponds to infinite path in T( X ). . . • Infinite path in a tree T is a sequence of vertices { v 0 , v 1 , v 2 , . . . } ⊂ T with v 0 = root and v i +1 is child of v i • We have bijection 1:1 → ¯ { Infinite paths in T( X ) } ← X (= p -adic closure of X ) Trees of definable sets in Z p 4 / 13 I. Halupczok

Examples of trees • Examples: p � Every node of T( X ) has p n children • X = Z n • X finite � each x ∈ X corresponds to infinite path in T( X ). . . • Infinite path in a tree T is a sequence of vertices { v 0 , v 1 , v 2 , . . . } ⊂ T with v 0 = root and v i +1 is child of v i • We have bijection 1:1 → ¯ { Infinite paths in T( X ) } ← X (= p -adic closure of X ) Trees of definable sets in Z p 4 / 13 I. Halupczok

Examples of trees • Examples: p � Every node of T( X ) has p n children • X = Z n • X finite � each x ∈ X corresponds to infinite path in T( X ). . . • Infinite path in a tree T is a sequence of vertices { v 0 , v 1 , v 2 , . . . } ⊂ T with v 0 = root and v i +1 is child of v i • We have bijection 1:1 → ¯ { Infinite paths in T( X ) } ← X (= p -adic closure of X ) Trees of definable sets in Z p 4 / 13 I. Halupczok

Examples of trees • Examples: p � Every node of T( X ) has p n children • X = Z n • X finite � each x ∈ X corresponds to infinite path in T( X ). . . • Infinite path in a tree T is a sequence of vertices { v 0 , v 1 , v 2 , . . . } ⊂ T with v 0 = root and v i +1 is child of v i • We have bijection 1:1 → ¯ { Infinite paths in T( X ) } ← X (= p -adic closure of X ) Trees of definable sets in Z p 4 / 13 I. Halupczok

Goal Question For which abstract trees T does there exist a definable X such that T ∼ = T( X ) ? • Obvious condition: T has no leaves. • Less obvious condition: • Suppose T ∼ = T( X ); fix infinite path P ⊂ T . • This yields x ∈ ¯ X . • Define: Side branch of P := subtree of T starting at a vertex of P , without P itself. ⇒ there exists x ′ ∈ X such • P has a side branch at depth λ ⇐ that v ( x ′ − x ) = λ • Thus: set of depths of side branches is definable subset of Z • Goal: Find combinatorial description of the set of (abstract) trees T( X ) for X definable. Trees of definable sets in Z p 5 / 13 I. Halupczok

Goal Question For which abstract trees T does there exist a definable X such that T ∼ = T( X ) ? • Obvious condition: T has no leaves. • Less obvious condition: • Suppose T ∼ = T( X ); fix infinite path P ⊂ T . • This yields x ∈ ¯ X . • Define: Side branch of P := subtree of T starting at a vertex of P , without P itself. ⇒ there exists x ′ ∈ X such • P has a side branch at depth λ ⇐ that v ( x ′ − x ) = λ • Thus: set of depths of side branches is definable subset of Z • Goal: Find combinatorial description of the set of (abstract) trees T( X ) for X definable. Trees of definable sets in Z p 5 / 13 I. Halupczok

Goal Question For which abstract trees T does there exist a definable X such that T ∼ = T( X ) ? • Obvious condition: T has no leaves. • Less obvious condition: • Suppose T ∼ = T( X ); fix infinite path P ⊂ T . • This yields x ∈ ¯ X . • Define: Side branch of P := subtree of T starting at a vertex of P , without P itself. ⇒ there exists x ′ ∈ X such • P has a side branch at depth λ ⇐ that v ( x ′ − x ) = λ • Thus: set of depths of side branches is definable subset of Z • Goal: Find combinatorial description of the set of (abstract) trees T( X ) for X definable. Trees of definable sets in Z p 5 / 13 I. Halupczok

Goal Question For which abstract trees T does there exist a definable X such that T ∼ = T( X ) ? • Obvious condition: T has no leaves. • Less obvious condition: • Suppose T ∼ = T( X ); fix infinite path P ⊂ T . • This yields x ∈ ¯ X . • Define: Side branch of P := subtree of T starting at a vertex of P , without P itself. ⇒ there exists x ′ ∈ X such • P has a side branch at depth λ ⇐ that v ( x ′ − x ) = λ • Thus: set of depths of side branches is definable subset of Z • Goal: Find combinatorial description of the set of (abstract) trees T( X ) for X definable. Trees of definable sets in Z p 5 / 13 I. Halupczok

Goal Question For which abstract trees T does there exist a definable X such that T ∼ = T( X ) ? • Obvious condition: T has no leaves. • Less obvious condition: • Suppose T ∼ = T( X ); fix infinite path P ⊂ T . • This yields x ∈ ¯ X . • Define: Side branch of P := subtree of T starting at a vertex of P , without P itself. ⇒ there exists x ′ ∈ X such • P has a side branch at depth λ ⇐ that v ( x ′ − x ) = λ • Thus: set of depths of side branches is definable subset of Z • Goal: Find combinatorial description of the set of (abstract) trees T( X ) for X definable. Trees of definable sets in Z p 5 / 13 I. Halupczok