The Poincaré series associated with finitely many monomial valuations

Abstract

Let $(R,M)$ be a Noetherian local domain, $F$ its quotient field and $v$ a valuation of $F$ centred at $R$. In the hope of understanding the valuation $v$, a useful tool is the so-called graded algebra relative to $v$, $gr_{v}R$. Its study can be understood as an attempt to approach the problem of classifying valuations. On the other hand, $gr_{v}R$ is also an object of interest since it can be provided with a geometrical meaning [15].