Growth of rank 1 valuation semigroups

TL;DR

This paper investigates the possible growth patterns of valuation semigroups of rank 1 dominating Noetherian local rings, providing bounds, examples, and refined estimates for their growth functions.

Contribution

It offers new precise growth estimates for rank 1 valuation semigroups and constructs examples with diverse growth rates, extending previous bounds.

Findings

Established polynomial bounds on the growth of valuation semigroups.

Constructed examples with growth rates like linear, rational power, and logarithmic.

Extended understanding of the possible behaviors of valuation semigroup growth functions.

Abstract

We consider the question of which semigroups can occur as the semigroup $S_R(\nu)$ 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 $\phi(n)=S_R(\nu)\cap (0,n)$ 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 $S_R(\nu)$ of a valuation dominating a Noetherian local ring. In the case of rank 1, it is possible to give more precise estimates of $\phi(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $\phi(n)$ on a regular local ring of dimension 2, such as $n({\alpha}$ for any rational $\alpha$ with $1\le\alpha\le 2$, and $n{log}(n)$.