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Characterizing Diophantine Henselian valuation rings and valuation ideals. (English) Zbl 1436.03192

The paper under review investigates definability of Henselian valuation rings and ideals in the language of rings. Most of the results in the paper deal with Diophantine (or existential) definability of these rings and ideals, but there are also examples of universal definability.
More specifically, given a field \(K\) and a Henselian valuation \(v\) of \(K\), the authors consider under what circumstances the valuation ring \(O_v\) and the valuation ideal \(m_v\) are definable over \(K\). The main result of the paper shows that existential definability of \(O_v\) and \(m_v\) depends exclusively on the residue field of the valuation. More precisely we have that the following statements are equivalent for a given field \(F\).
\(O_v\) (resp. \(m_v\)) are existentially definable (in the language of rings without parameters) in some field \(K\) for some equicharacteristic Henselian non-trivially valued field \((K,v)\) with residue field \(F\).
\(O_v\) (resp. \(m_v\)) are existentially definable (in the language of rings without parameters) in all Henselian fields \((K,v)\) with residue field elementarily equivalent to \(F\).
There is no elementary expansion \(F \preceq F^*\) with a non-trivial valuation \(v\) on \(F^*\) for which the residue field \(F^*v\) embeds into \(F^*\) (resp. with a non-trivial Henselian valuation \(v\) on a subfield \(E\) of \(F^*\) with \(Ev \cong F^*\)).

This remarkable result produces new easier proofs of many old results as well as an impressive number of the new results. One of these new results that this reviewer found particularly interesting states the following:
Let \(F\) be a field. Then there exists an existential definition of the maximal ideal in \(F((t))\) if and only if \(F\) is not a large field.

MSC:

03C60 Model-theoretic algebra
03C40 Interpolation, preservation, definability
11D88 \(p\)-adic and power series fields
11U09 Model theory (number-theoretic aspects)
12J10 Valued fields
12L12 Model theory of fields
14G05 Rational points
14G20 Local ground fields in algebraic geometry
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