On the implicit constant fields and key polynomials for valuation algebraic extensions

Abstract

This article is a natural continuation of our previous works [7] and [6]. In this article, we employ similar ideas as in [4] to provide an estimate of IC(K(X)|K, v) when (K(X)|K, v) is a valuation algebraic extension. Our central result is an analogue of [6, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.

Abstract Key Polynomials and MacLane-Vaquié chains

“…We now recall the construction of a pseudo-convergent sequence of transcendental type given by Dutta (see Section 5, [2]). Remark 3.11.…”

“…With the help of the above sequence Dutta gave a complete sequence of ABKPs for w. Theorem 3.12 (Theorem 7.4, [2]). Assume that w is a valuation-algebraic extension of v to K(X).…”