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Cell decomposition in P-minimal structures
Saskia Chambille
Joint work with Pablo Cubides Kovacsics & Eva Leenknegt
Departement of Mathematics
KU Leuven
Konstanz, July 22, 2016
Saskia Chambille
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
1 / 11
Tree pictures of the p-adic numbers
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0
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
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ord(a) ∈ Z
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0
Saskia Chambille
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
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a
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ord(a) ∈ Z
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1
2
1
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0 2
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Saskia Chambille
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
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a
ord(a) + 2
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ac2 (a)
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ord(a) ∈ Z
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0 2
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Saskia Chambille
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
0
a
ord(a) + 2
0
ac2 (a)
0
ord(a) ∈ Z
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0 2
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
a
a
0
Saskia Chambille
ord(a) + 2
0
ord(a) + 2
ord(a) ∈ Z
ord(a)
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Tree pictures of the p-adic numbers
a
a
0
Saskia Chambille
ord(a) + 2
0
ord(a) + 2
ord(a) ∈ Z
ord(a)
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
2 / 11
Definitions
Let p bePa prime number. Qp is the field of the p-adic numbers.
∞
i
Q×
p = { i=k ai p | k ∈ Z, ai ∈ {0, 1, . . . , p − 1}, ak 6= 0}.
Saskia Chambille
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
3 / 11
Definitions
Let p bePa prime number. Qp is the field of the p-adic numbers.
∞
i
Q×
p = { i=k ai p | k ∈ Z, ai ∈ {0, 1, . . . , p − 1}, ak 6= 0}.
(
P
i
k
if a = ∞
i=k ai p ;
ord : Qp → Z ∪ {∞} : a 7→
∞ if a = 0.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
3 / 11
Definitions
Let p bePa prime number. Qp is the field of the p-adic numbers.
∞
i
Q×
p = { i=k ai p | k ∈ Z, ai ∈ {0, 1, . . . , p − 1}, ak 6= 0}.
(
P
i
k
if a = ∞
i=k ai p ;
ord : Qp → Z ∪ {∞} : a 7→
∞ if a = 0.
For each m ∈ N0 ,
m
(P
m−1
acm : Qp → Zp /p Zp : a 7→
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i=0
ai+k p i
0
Cell decomposition in P-minimal structures
P
i
if a = ∞
i=k ai p ;
if a = 0.
Konstanz, July 22, 2016
3 / 11
Definitions
Let p bePa prime number. Qp is the field of the p-adic numbers.
∞
i
Q×
p = { i=k ai p | k ∈ Z, ai ∈ {0, 1, . . . , p − 1}, ak 6= 0}.
(
P
i
k
if a = ∞
i=k ai p ;
ord : Qp → Z ∪ {∞} : a 7→
∞ if a = 0.
For each m ∈ N0 ,
m
(P
m−1
acm : Qp → Zp /p Zp : a 7→
i=0
ai+k p i
0
P
i
if a = ∞
i=k ai p ;
if a = 0.
Clopen balls Bγ (a) := {x ∈ Qp | ord(x − a) ≥ γ}, γ ∈ Z.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
3 / 11
Semialgebraic sets
Let K be a field and Lring = {+, −, ·, 0, 1}.
Definition
The semialgebraic sets in K are de Lring -definable subsets of K .
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
4 / 11
Semialgebraic sets
Let K be a field and Lring = {+, −, ·, 0, 1}.
Definition
The semialgebraic sets in K are de Lring -definable subsets of K .
o-minimality
K is a real closed field if it is
Lring -elementary equivalent to
R.
x < y ⇔ ∃z(x + z 2 = y ).
Lor := Lring ∪ {<}.
(K , Lor ) has QE.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
4 / 11
Semialgebraic sets
Let K be a field and Lring = {+, −, ·, 0, 1}.
Definition
The semialgebraic sets in K are de Lring -definable subsets of K .
P-minimality
o-minimality
K is a real closed field if it is
Lring -elementary equivalent to
R.
K is a p-adically closed field if
it is Lring -elementary equivalent
to a finite field extension of Qp .
x < y ⇔ ∃z(x + z 2 = y ).
x ∈ Pk ⇔ ∃y 6= 0(y k = x).
Lor := Lring ∪ {<}.
LMac := Lring ∪ {Pk }k>0 .
(K , Lor ) has QE.
(K , LMac ) has QE. (Prestel &
Roquette, ‘84)
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
4 / 11
Examples of semialgebraic sets
X ⊆ R semialgebraic ⇒ X is a Boolean combination of sets of the form
{x ∈ R | f (x) = 0}
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{x ∈ R | g (x) > 0}.
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
5 / 11
Examples of semialgebraic sets
X ⊆ R semialgebraic ⇒ X is a Boolean combination of sets of the form
{x ∈ R | f (x) = 0}
{x ∈ R | g (x) > 0}.
X ⊆ Qp semialgebraic ⇒ X is a Boolean combination of sets of the form
{x ∈ Qp | f (x) = 0}
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{x ∈ Qp | g (x) ∈ Pk }.
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
5 / 11
Examples of semialgebraic sets
X ⊆ R semialgebraic ⇒ X is a Boolean combination of sets of the form
{x ∈ R | f (x) = 0}
{x ∈ R | g (x) > 0}.
X ⊆ Qp semialgebraic ⇒ X is a Boolean combination of sets of the form
{x ∈ Qp | f (x) = 0}
{x ∈ Qp | g (x) ∈ Pk }.
0
Instead of (Pk )k>0 we prefer (Qm,n )m,n>0 , where
Qm,n := {x ∈ Q×
p | ord(x) ≡ 0 mod n, acm (x) = 1}.
m
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Cell decomposition in P-minimal structures
n
Konstanz, July 22, 2016
5 / 11
o-minimality and P-minimality
Definition
Let L ⊇ Lring and let K be a real closed field. The structure (K , L) is
o-minimal if every L-definable subset X ⊆ K is also Lring -definable.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
6 / 11
o-minimality and P-minimality
Definition
Let L ⊇ Lring and let K be a real closed field. The structure (K , L) is
o-minimal if every L-definable subset X ⊆ K is also Lring -definable.
Definition
Let L ⊇ Lring and let K be a p-adically closed field. The structure (K , L)
is P-minimal if for every (K 0 , L) ≡ (K , L), every L-definable subset
X ⊆ K 0 is also Lring -definable.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
6 / 11
Classical cells
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
σ
(0)-cell.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
σ
σ
m
(1)-cell.
(0)-cell.
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n
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
σ
σ
ord(β)
m
(1)-cell.
(0)-cell.
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n
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
σ
σ
m
n
ord(α)
(1)-cell.
(0)-cell.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
σ
σ
ord(β)
m
n
ord(α)
(1)-cell.
(0)-cell.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
7 / 11
Classical cells
C=
ord(α(s))1 ord(x − σ(s))2 ord(β(s)),
(s, x) ∈ S × K x − σ(s) ∈ λQm,n
S is a definable set, α, β : S → K × and σ : S → K are definable functions,
i can be either < or ‘no condition’, n, m ∈ N0 and λ ∈ K .
We call σ the center of the cell.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
8 / 11
Classical cells
C=
ord(α(s))1 ord(x − σ(s))2 ord(β(s)),
(s, x) ∈ S × K x − σ(s) ∈ λQm,n
S is a definable set, α, β : S → K × and σ : S → K are definable functions,
i can be either < or ‘no condition’, n, m ∈ N0 and λ ∈ K .
We call σ the center of the cell.
σ(s)
σ(s)
ord(β(s))
m
acm (λ)
n
ord(α(s))
λ = 0.
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λ 6= 0 and 1 = 2 =<.
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
8 / 11
Definable Skolem functions
Definition
A structure (K , L) has definable Skolem functions if for every definable
X ⊆ K r +1 there exists a definable section σ : π(X ) → K . This means that
(x, σ(x)) ∈ X for all x ∈ π(X ). By π : K r +1 → K r we mean the
projection onto the first r coordinates.
Theorem (Mourgues, ‘09)
A P-minimal structure (K , L) has cell decomposition (with classical cells)
if and only if it has definable Skolem functions.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
9 / 11
Definable Skolem functions
Definition
A structure (K , L) has definable Skolem functions if for every definable
X ⊆ K r +1 there exists a definable section σ : π(X ) → K . This means that
(x, σ(x)) ∈ X for all x ∈ π(X ). By π : K r +1 → K r we mean the
projection onto the first r coordinates.
Theorem (Mourgues, ‘09)
A P-minimal structure (K , L) has cell decomposition (with classical cells)
if and only if it has definable Skolem functions.
Remarks:
All o-minimal structures (K , L) have definable Skolem functions (and
cell decomposition).
Cubides and Nguyen have found a P-minimal structure that does not
have definable Skolem functions!
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
9 / 11
Centers of cells
Cell condition: C (s, y , x) =
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ord(α(s))1 ord(x − y )2 ord(β(s))
∧(x − y ) ∈ λQm,n ∧ s ∈ S.
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
10 / 11
Centers of cells
Cell condition: C (s, y , x) =
ord(α(s))1 ord(x − y )2 ord(β(s))
∧(x − y ) ∈ λQm,n ∧ s ∈ S.
For any function σ : S → K ,
we define
σ(s)
ord(β(s))
C σ := {(s, x) ∈ S×K | C (s, σ(s), x)}.
n
ord(α(s))
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
10 / 11
Centers of cells
Cell condition: C (s, y , x) =
For any function σ : S → K ,
we define
σ(s) σ 0 (s)
m
ord(α(s))1 ord(x − y )2 ord(β(s))
∧(x − y ) ∈ λQm,n ∧ s ∈ S.
ord(β(s))
ρmax (s)
C σ := {(s, x) ∈ S×K | C (s, σ(s), x)}.
0
We have C σ = C σ .
n
ord(α(s))
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
10 / 11
Centers of cells
Cell condition: C (s, y , x) =
For any function σ : S → K ,
we define
σ(s) σ 0 (s)
m
ord(β(s))
ρmax (s)
C σ := {(s, x) ∈ S×K | C (s, σ(s), x)}.
0
We have C σ = C σ .
The possible centers are the
sections of
[
Σ :=
{s}×Bρmax (s)+m (σ(s)) ⊆ S×K .
n
s∈S
ord(α(s))
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ord(α(s))1 ord(x − y )2 ord(β(s))
∧(x − y ) ∈ λQm,n ∧ s ∈ S.
Then C σ = {(s, x) | ∃σ 0 ∈ Σs C (s, σ 0 , x)}.
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
10 / 11
Inseperably clustered cells
σ1 (s)
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σ2 (s)
σ3 (s)
X = C σ1 t . . . t C σk .
Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
11 / 11
Inseperably clustered cells
σ1 (s)
σ2 (s)
σ3 (s)
X = C σ1 t . . . t C σk .
There exists a definable Σ ⊆ S × K
for which each Σs is a union of k
balls of the same size and such that
X = {(s, x) | ∃σ 0 ∈ Σs C (s, σ 0 , x)}.
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Cell decomposition in P-minimal structures
Konstanz, July 22, 2016
11 / 11