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- W2949409411 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 $ninNN$, 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 $1lealphale 2$, and $n{log}(n)$." @default.
- W2949409411 created "2019-06-27" @default.
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- W2949409411 date "2008-09-20" @default.
- W2949409411 modified "2023-09-27" @default.
- W2949409411 title "Growth of rank 1 valuation semigroups" @default.
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