Key polynomials and distinguished pairs

On chains associated with abstract key polynomials

“…In the next result we give a connection between saturated distinguished chains and Okutsu frames. It may be pointed that a similar result is also proved in Theorem 2.6 of [13] and Corollary 3.5 of [7] but our proof is elementary.…”

“…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.…”

Abstract key polynomials and distinguished pairs

“…Theorem 1.15 (Theorem 3.1, [10]). Let (K, v) be a henselian valued of arbitrary rank and F be a key polynomial over a residually transcendental extension w of v to K(X) defined by a (K, v)-minimal pair (α, δ).…”

“…[1]- [4]) to henselian valued fields. The notion of saturated distinguished chains is used to find results about irreducible polynomials and to obtain various invariants associated with elements of K. Recently Jakhar and Sangwan in [10], established a connection between distinguished pairs and key polynomials over a residually transcendental extension of v. In this paper, we associate distinguished pairs with abstract key polynomials over an extension w of v to K(X). Moreover, it will be shown that abstract key polynomials for w leads to saturated distinguished chains.…”