  Given a noetherian local domain $R$ and a valuation $\nu$ of its field of
fractions which is non negative on $R$, we derive some very general bounds on
the growth of the number of distinct valuation ideals of $R$ corresponding to
values lying in certain parts of the value group $\Gamma$ of $\nu$. We show
that this growth condition imposes restrictions on the semigroups
$\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger that those
resulting from the previous paper \cite{C2} of the first author. Given an
ordered embedding $\Gamma\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$
is the rank of $\nu$, we also study the shape in ${\mathbf R}^h$ of the parts
of $\Gamma$ which appear naturally in this study. We give examples which show
that this shape can be quite wild in a way which does not depend on the
embedding and suggest that it is a good indicator of the complexity of the
semigroup $\nu(R\setminus \{0\})$.

Complex Variables;Commutative Algebra;Algebraic Geometry