Generating sequences of rank 1 valuations

TL;DR

This paper presents an algorithm for constructing generating sequences of rank 1 valuations in 3-dimensional regular local rings, generalizing previous methods and including explicit examples for higher-rank valuations.

Contribution

It introduces a new algorithm for generating sequences of rank 1 valuations in three-dimensional regular local rings, extending prior work to non-rational rank cases.

Findings

Algorithm successfully constructs generating sequences for rank 1 valuations.

Explicit example of a rank 3 valuation with key polynomials.

Comparison demonstrates the algorithm's applicability to higher-rank valuations.

Abstract

We consider a rank 1 valuation $\nu$ centered in a regular 3-dimensional local ring $R$. We assume that the residue field $k$ of $\nu$ is contained in $R$. An algorithm for constructing a generating sequences for $\nu$ in $R$ is provided. This is a generalization of our previous algorithm where only valuations of rational rank 1 were considered. We then give a comparison example of a rational rank 3 valuation centered in a ring of polynomials in three variables for which we explicitly compute a set of key polynomials and a generating sequence.