Growth of rank 1 valuation semigroups

Cutkosky, Steven Dale; Dalili, Kia; Kashcheyeva, Olga

Growth of rank 1 valuation semigroups

Authors

Cutkosky, Steven Dale

Dalili, Kia

Kashcheyeva, Olga

Description

We consider the question of which semigroups can occur as the semigroup $S_R(ν)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $ϕ(n)=S_R(ν)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of rank 1 well ordered semigroups which are not the value semigroup $S_R(ν)$ of a valuation dominating a Noetherian local ring. In the case of rank 1, it is possible to give more precise estimates of $ϕ(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $ϕ(n)$ on a regular local ring of dimension 2, such as $n(α$ for any rational $α$ with $1\leα\le 2$, and $n{log}(n)$.
13 pages