In this paper, for a valued field [Formula: see text] of arbitrary rank and an extension [Formula: see text] of [Formula: see text] to [Formula: see text] a relation between induced complete sequences of abstract key polynomials and MacLane-Vaquié chains is given.

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In this paper, for a henselian valued field [Formula: see text] of arbitrary rank and an extension [Formula: see text] of [Formula: see text] to [Formula: see text] we use abstract key polynomials for [Formula: see text] to obtain distinguished pairs and saturated distinguished chains.

For a nontrivial approximation type [Formula: see text] over a valued field [Formula: see text] it is known that there is an extension [Formula: see text] of [Formula: see text] to [Formula: see text] such that [Formula: see text] is realized by [Formula: see text] in [Formula: see text] In this paper, we show that [Formula: see text] is determined by the first step [Formula: see text] of any MacLane–Vaquié chain of [Formula: see text] Also, the properties of [Formula: see text] as an approximation type are reflected in the valuative properties of [Formula: see text] and [Formula: see text]

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