Growth of Rank 1 Valuation Semigroups

Abstract

We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.

Key Words:

• Valuation semigroup

2000 Mathematics Subject Classification:

• Primary 13A18, 14E15
• Secondary 06F05

ACKNOWLEDGMENT

The first author was partially supported by NSF.

Notes

Communicated by I. Swanson.