Abstract: 

In this talk, we will address a question of  Krapp, Kuhlmann and Link: can we characterise the field k of characteristic 0, which has the following property: in all Henselian valued fields with residue k, the valuation ring is definable in the language of rings? Similarly, one can ask how to characterise the value group G which has the following property: in all Henselian valued fields of equicharacteristic 0 with value group G, the valuation ring is definable in the language of ring. We will see an answer for both questions, that we formulate as a (relatively simple) Ax-Kochen-Ershov principle for the definability of the valuation. This will lead us to interesting questions about ordered abelian groups and linear orders. This is a joint work with Blaise Boissonneau, Franziska Jahnke and Anna De Mase.

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