Constructing examples of semigroups of 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.