Key polynomials over valued fields

Abstract

Let K be a field. For any valuation µ on K[x] admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of µ-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of µ.