Growth of rank 1 valuation semigroups

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We consider the question of which semigroups can occur as the semigroup Sᵣ(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)=Sᵣ(ν)∩ (0,n) for n∈, starting with the upper bound of Pᵣ(n), where Pᵣ(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 Sᵣ(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≤α≤ 2, and nlog(n).