On the existential theory of equicharacteristic henselian valued fields

Arno Fehm (Universit├Ąt Konstanz)

Frank Adams 1,

The first order theory of a henselian valued field (K,v) of residue characteristic zero is well-understood through the celebrated Ax-Kochen-Ershov principle, which states that it is completely determined by the theory of the residue field and the theory of the value group. For henselian valued fields of positive residue characteristic, no such general principle is known. I will report on joint work with Will Anscombe in which we study the theory of equicharacteristic henselian valued fields of arbitrary characteristic and prove an Ax-Kochen-Ershov principle for existential sentences. I will also discuss applications to the result of Denef-Schoutens on the existential decidability (Hilbert's tenth problem) of the local field F_q((t)).

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