Extension d'une valuation

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Valuations in algebraic field extensions (2006, methodology): “Two main techniques used in this paper are higher Newton polygons and a version of MacLane’s key polynomials, similar to those considered by M. Vaqui´e ([9], [ 10 ], [11], and [12]), and reminiscent of related objects studied by Abhyankar and Moh (approximate roots [1], [2]) and T.C.”
Key polynomials and pseudo-convergent sequences (2016, background): “Vaquié developed a more general theory of key polynomials (see [9]), which extends the results of MacLane for a general valued field (that is, the given valuation of K is no longer assumed to be discrete).”
On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop (2012, methodology): “Key polynomials were developed by Ostrowski (see [71]) and MacLane (see [54]) in special cases and in full generality by Vaquié (see [83]-[87]) to determine all the extensions of a given valuation of the field K to the field K(y), where y may be algebraic over K or not.”
Dynamics on Berkovich Spaces in Low Dimensions (2012, methodology): “acLane [Mac36]. While beautiful, this method is coordinate dependent and involves some quite delicate combinatorics. In addition, even though there is a notion of key polynomials in higher dimensions [Vaq07], these seem hard to use for our purposes. The geometric approach, using blowups and dual gra”