Constructing examples of semigroups of valuations

TL;DR

This paper develops algorithms and methods to construct and analyze valuations and their semigroups in three-variable function fields, providing explicit examples and insights into their structure.

Contribution

It introduces an algorithm for generating sequences of valuations in regular local rings and a method for determining valuations on three-variable rational function fields.

Findings

Constructed explicit examples of valuations in three variables.

Developed an algorithm for generating valuation sequences.

Analyzed the semigroups of values for these valuations.

Abstract

We work with rational rank 1 valuations centered in regular local rings. Given an algebraic function field $K$ of transcendence degree 3 over $k$, a regular local ring $R$ with $QF(R)=K$ and a $k$-valuation $\nu$ of $K$, we provide an algorithm for constructing a generating sequences for $\nu$ in $R$. We then develop a method for determining a valuation $\nu$ on $k(x,y,z)$ through the sequence of defining values. Using the above results we construct examples of valuations centered in $k[x,y,z]_{(x,y,z)}$ and investigate their semigroups of values.