Key polynomials for simple extensions of valued fields

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.