Limit key polynomials as p-polynomials

Abstract

The main goal of this paper is to characterize limit key polynomials for a valuation ν on K[x]. We consider the set Ψα of key polynomials for ν of degree α. We set p to be the exponent characteristic of ν. Our first main result (Theorem 1.1) is that if F is a limit key polynomial for Ψα, then the degree of F is prα for some r∈N. Moreover, in Theorem 1.2, we show that there exist Q∈Ψα and F a limit key polynomial for Ψα, such that the Q-expansion of F only has terms which are powers of p.