Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We discuss definability of henselian valuation rings in the Macintyre language L Mac , the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃‐∅‐definable in L Mac , and henselian valuation rings with value group double-struckZ are uniformly ∃∀‐∅‐definable in the ring language, but not uniformly ∃‐∅‐definable in L Mac . We apply these results to local fields Qp and double-struckFp((t)), as well as to higher dimensional local fields.

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“…. , in K for any p-henselian valued field (K, v) with discrete p-regular value group satisfying one of the three conditions (1)- (3).…”

Section: Value Group Z In the Ring Languagementioning

confidence: 99%

“…[17]. In this language, the following definition has recently been obtained in [3,Theorem 3] using results from the model theory of pseudo-finite fields. Theorem 1.1 (Cluckers-Derakhshan-Leenknegt-Macintyre).…”

Section: Introductionmentioning

confidence: 99%

Uniform definability of henselian valuation rings in the Macintyre language:

Fehm

Prestel

2015

Bull. London Math. Soc.

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“…The special case where K is a finite extension of Q p was proven by Cluckers, Derakhshan, Leenknegt, and Macintyre in [5,Theorem 6]. §3.…”

Section: Theorem 12 For Every > 0 There Exists An ∃-∅-Formula ϕ Andmentioning

confidence: 99%

“…For details the reader may consult [5,Theorem 4], where it is shown that no ∃-∅-formula can define the valuation ring uniformly for all finite extensions of a fixed henselian valued field K.…”

Section: Corollary 36 Let K Be a Henselian Valued Field With Valuatmentioning

confidence: 99%

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Existential ∅-Definability of Henselian Valuation Rings

Fehm¹

2015

J. symb. log.

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Abstract. In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F [[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. §1. Introduction. The question of first order definability of valuation rings in their quotient fields has a long history. Given a valued field K, one is interested in whether there exists a first order formula ϕ in the language L = {+, −, ·, 0, 1} of rings such that the set ϕ(K) defined by ϕ in K is precisely the valuation ring, and what complexity such formula must have.Many results of this kind are known for henselian valued fields, like fields of formal power series K = F ((t)) over a field F , and their valuation ring F [[t]]. In this setting, a definition going back to Julia Robinson gives an existential definition of the valuation ring using the parameter t. Later, Ax [2] gave a definition of the valuation ring, which uses no parameters, but is not existential.Recently, Anscombe and Koenigsmann [1] succeeded to give an existential and parameter-free definition of F [[t]] in F ((t)) in the special case where F = F q is a finite field. Their proof uses the fact that F q can be defined in F q ((t)) by the quantifier-free formula x q − x = 0. In particular, their result does not apply to any infinite field F , and their formula depends heavily on q.In this note we simplify and extend their method. As a first application we get the following general definability result for henselian valued fields with finite or pseudo-algebraically closed residue fields (Theorem 2.6 and Theorem 3.5), which generalizes [1, Theorem 1.1] on F q ((t)) and [5, Theorem 6] As a further application, in Section 4, we find definitions of the valuation ring which are uniform for large (infinite) families of finite residue fields, like the following one for finite prime fields (Theorem 4.3):

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