The Henselian defect for valued function fields

Abstract

The notion of defect for finite algebraic extensions of valued fields is classical and due to Ostrowski. Recently Matignon has generalized Ostrowski’s definition to <italic>rk</italic> 1 (residually transcendental) valued function fields and used it to prove a very sharp version of the genus reduction inequality for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus dimension"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>dim</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 - \dim</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function fields. The further generalization of the notion of defect to valued function fields of arbitrary <italic>rk</italic> is treated here.