Eliminating field quantifiers in strongly dependent henselian fields

Abstract

We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper K comma v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(K,v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strongly dependent, then so is its henselization.