Defectless polynomials over henselian fields and inductive valuations

MacLane-Vaquié chains of valuations on a polynomial ring

“…This result is proved in [5,Thm. 3.9] for the group Λ = Γ Q , but the proof is valid for any ordered group Λ.…”

On chains associated with abstract key polynomials

“…Remark 2.18. It can be shown that the above definition of an Okutsu frame is equivalent to the one given in [12].…”

“…The notion of Okutsu frame was introduced by Okutsu in 1982 for local fields [20], and then studied by Nart in [6], which was later generalized to henselian valued field of arbitrary rank in [12] and [13].…”

“…If w is any commensurable extension of v to K(X), then from the above embedding we have w −∞ < w, and w −∞ is called the minimal extension of v to K(X). Moreover the augmentation of w −∞ with respect to the key polynomial φ 0 = X + a for some a ∈ K, defined in a natural way, is denoted by w 0 (see [12,Subsection 2.2]). Definition 2.26.…”

“…In 1982, Okutsu associated to a monic irreducible polynomial F ∈ K[X] a family of monic irreducible polynomials, F 1 , ..., F r , called the primitive divisor polynomials of F [20], later these polynomials were studied in papers [6], [7], [12] and [13], and they called the chain of such polynomials [F 1 , ..., F r ], an Okutsu frame for F. Moreover, they proved that Okutsu frames, saturated distinguished chains and optimal Maclane chains are closely related. In this paper, we also establish a similar connection between saturated distinguished chains and Okutsu frames, however, our proof is elementary.…”