Imaginaries and invariant types in existentially closed valued differential fields

Rideau, Silvain

doi:10.1515/crelle-2016-0036

Cited by 11 publications

(20 citation statements)

References 21 publications

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“…Corollary shows that the hypothesis of Proposition holds, and therefore provides us with an

\overset{A}{true}

‐definable (in

\overset{T}{true}

) global

T

‐type

p

, consistent with

boldX

. Now, by [, Corollary A.7], the model provided by Proposition satisfies the condition in Proposition , and Corollary shows that

T_{normalP, prefixval}^{★}

(hence

\overset{T}{true}

) is complete, so Proposition applies to show that

p

is definable in

T

over

bold-italic bold-scriptG (\hat{A}) = A

□

…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 89%

“…We begin with a review of some results from . Consider an arbitrary complete theory

T

, and a fixed universal domain

normalM

.…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 99%

“…This is shown in several steps. The following result from produces an

ACVF

‐type consistent with a given definable set in a suitable expansion, definable in that expansion: Proposition Let

\hat{T} \supseteq {ACVF}^{scriptG}

be a theory in a countable language

\overset{L}{true}

, such that (1)

\overset{T}{true}

eliminates imaginaries; (2)

boldk

and

normalΓ

are stably embedded; (3)the induced theories on

boldk

and

{normalΓ}^{eq}

eliminate

\exists^{\infty}

. Let

\hat{A} = {acl}_{\overset{T}{true}} false(\overset{A}{true} false) \subseteq \hat{N} ⊧ \hat{T}

and let

boldX

be a non‐empty strict pro‐

\overset{A}{true}

‐definable set of

boldVF

elements. Then there exists a global

{ACVF}^{G}

‐type

p

consistent with

boldX

, which is

\overset{A}{true}

‐definable (in

\overset{T}{true}

).…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 99%

“…Since then, it has been shown that various theories of valued fields eliminate imaginaries down to the geometric sorts, namely the theory RCVF of real closed valued fields (by work of Mellor ), the theories of

p

‐adic fields and of ultraproducts of

p

‐adic fields (by work of Hrushovski, Martin and the third author ) and the theory

{VDF}_{EC}

of existentially closed valued differential fields

(K, v, \partial)

of residue characteristic 0 satisfying

v (\partial (x)) ⩾ v (x)

for all

x

(by work of the third author ).…”

Section: Introductionmentioning

confidence: 99%

“…The first proof consists in finding a canonical way of representing a semi‐algebraic set definable in a separably closed valued field

K

by the

K

‐points of a set definable in

\bar{K} ⊧ {ACVF}_{p, p}

. The second proof, inspired by the work of the third author on the theory

{VDF}_{EC}

, is much more local. We only achieve a correspondence between

K

and

\bar{K}

at the level of types.…”

Section: Introductionmentioning

confidence: 99%

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Imaginaries in separably closed valued fields

Hils

Kamensky

Rideau

2018

Proc. London Math. Soc.

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We show that separably closed valued fields of finite imperfection degree (either with λ‐functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite imperfection degree are metastable and that the space of stably dominated types is strict pro‐definable.

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“…Corollary shows that the hypothesis of Proposition holds, and therefore provides us with an

\overset{A}{true}

‐definable (in

\overset{T}{true}

) global

T

‐type

p

, consistent with

boldX

. Now, by [, Corollary A.7], the model provided by Proposition satisfies the condition in Proposition , and Corollary shows that

T_{normalP, prefixval}^{★}

(hence

\overset{T}{true}

) is complete, so Proposition applies to show that

p

is definable in

T

over

bold-italic bold-scriptG (\hat{A}) = A

□

…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 89%

“…We begin with a review of some results from . Consider an arbitrary complete theory

T

, and a fixed universal domain

normalM

.…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 99%

“…This is shown in several steps. The following result from produces an

ACVF

‐type consistent with a given definable set in a suitable expansion, definable in that expansion: Proposition Let

\hat{T} \supseteq {ACVF}^{scriptG}

be a theory in a countable language

\overset{L}{true}

, such that (1)

\overset{T}{true}

eliminates imaginaries; (2)

boldk

and

normalΓ

are stably embedded; (3)the induced theories on

boldk

and

{normalΓ}^{eq}

eliminate

\exists^{\infty}

. Let

\hat{A} = {acl}_{\overset{T}{true}} false(\overset{A}{true} false) \subseteq \hat{N} ⊧ \hat{T}

and let

boldX

be a non‐empty strict pro‐

\overset{A}{true}

‐definable set of

boldVF

elements. Then there exists a global

{ACVF}^{G}

‐type

p

consistent with

boldX

, which is

\overset{A}{true}

‐definable (in

\overset{T}{true}

).…”

Section: Imaginaries Definable Types and Dense Pairsmentioning

confidence: 99%

p

‐adic fields and of ultraproducts of

p

‐adic fields (by work of Hrushovski, Martin and the third author ) and the theory

{VDF}_{EC}

of existentially closed valued differential fields

(K, v, \partial)

of residue characteristic 0 satisfying

v (\partial (x)) ⩾ v (x)

for all

x

(by work of the third author ).…”

Section: Introductionmentioning

confidence: 99%

“…The first proof consists in finding a canonical way of representing a semi‐algebraic set definable in a separably closed valued field

K

by the

K

‐points of a set definable in

\bar{K} ⊧ {ACVF}_{p, p}

. The second proof, inspired by the work of the third author on the theory

{VDF}_{EC}

, is much more local. We only achieve a correspondence between

K

and

\bar{K}

at the level of types.…”

Section: Introductionmentioning

confidence: 99%

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Imaginaries in separably closed valued fields

Hils

Kamensky

Rideau

2018

Proc. London Math. Soc.

Self Cite

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show abstract

Some definable types that cannot be amalgamated

Hils

Mennuni

2023

Mathematical Logic Qtrly

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show abstract

Topological fields with a generic derivation

Kovacsics

Point

2023

Annals of Pure and Applied Logic

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Imaginaries and invariant types in existentially closed valued differential fields

Cited by 11 publications

References 21 publications

Imaginaries in separably closed valued fields

Imaginaries in separably closed valued fields

Some definable types that cannot be amalgamated

Topological fields with a generic derivation

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