Key polynomials for simple extensions of valued fields

Francisco Javier Herrera Govantes (1) , Wael Mahboub , Miguel Angel Olalla Acosta (1) , Mark Spivakovsky (2)

1 DALG-US - Departamento de Algebra [Sevilla]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper we present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaquié, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Let (K, ν_0) be a valued field. Given a simple transcendental extension of valued fields ι : K → K(x) we associate to ι a countable well ordered set of polynomials of K[x] called key polynomials. We define limit key polynomials and give an explicit description of them. We show that the order type of the set of key polynomials is bounded by ω × ω. If char k_ν_0 = 0 and rk ν_0 = 1, the order type is bounded by ω + 1.

Mots clés: valuations; key polynomials; the extension problem

Domaines: Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]

Francisco Javier Herrera Govantes, Wael Mahboub, Miguel Angel Olalla Acosta, Mark Spivakovsky. Key polynomials for simple extensions of valued fields. Journal of Singularities, 2022, 25, pp.197-267. ⟨10.5427/jsing.2022.25k⟩. ⟨hal-01887698v2⟩