Limit key polynomials as $p$-polynomials

TL;DR

This paper characterizes limit key polynomials for valuations on polynomial rings, showing their degrees are powers of the characteristic exponent and that their expansions involve only powers of this prime.

Contribution

It provides a new characterization of limit key polynomials, linking their degrees and expansions to the prime characteristic of the valuation.

Findings

Limit key polynomials have degrees of the form p^r * alpha.

Existence of limit key polynomials with expansions involving only powers of p.

Degree of limit key polynomials is determined by the prime characteristic.

Abstract

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