Approximate AKE principle completely describes elementary equivalence for metric valued fields in equicharacteristic 0 via residue field and value group; metric valued difference fields have no model companion.
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An Approximate AKE Principle for Metric Valued Fields
Approximate AKE principle completely describes elementary equivalence for metric valued fields in equicharacteristic 0 via residue field and value group; metric valued difference fields have no model companion.