Minimal limit key polynomials

TL;DR

This paper extends the theory of minimal limit key polynomials for valuations on polynomial rings, applying ordered abelian group cuts to generalize results across different valuation ranks and exploring properties in unbounded cases.

Contribution

It introduces a unified approach using cuts on ordered abelian groups to extend minimal limit key polynomial theory to arbitrary rank valuations.

Findings

Extended results to vertically bounded sets of key polynomials for arbitrary rank valuations

Analyzed properties of minimal limit key polynomials in unbounded cases

Unified theory using cuts on ordered abelian groups

Abstract

In this paper, we extend the theory of minimal limit key polynomials of valuations on the polynomial ring $\kx$. We use the theory of cuts on ordered abelian groups to show that the previous results on bounded sets of key polynomials of rank-one valuations, extend to vertically bounded sets of key polynomials of valuations of an arbitrary rank. We discuss as well properties of minimal limit key polynomials in the vertically unbounded case.