Key polynomials for simple extensions of valued fields

Abstract

In this paper we present a refined version of MacLane's theory of key polynomials [16]-[17], similar to those considered by M. Vaquié [24]-[27], and reminiscent of related objects studied by Abhyankar and Moh (approximate roots [1], [2]) and T.C. Kuo [14], [15].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