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A measure of integrity for local analytic algebras. (English) Zbl 0587.32016

The author is interested in three types of order for f in an analytic local ring \({\mathcal O}={\mathbb{C}}\{x_ 1,...,x_ n\}/I\) with maximal ideal \({\mathcal M}\). These orders are the algebraic order, \(\nu (f):=\sup \{p: f\in {\mathcal M}^ p\};\) the reduced order, \({\bar \nu}(f):=\lim_{k\to \infty} \nu (f^ k)/k;\) and the analytic order along a subanalytic set A, \(\mu_ A(f):=\sup \{p: | \tilde f(x)| \leq \alpha | x|^ p\) on \(U\cap A\) for some \(\alpha\), open set U, and representative \(\tilde f\}\). Certain inequalities are obvious; \({\bar \nu}(f) \geq \nu (f),\) \(\nu (fg) \geq \nu (f) + \nu (g),\) \(\nu (f\circ \Phi) > \nu (f)(\Phi: {\mathbb{C}}_ 0^ n\to {\mathbb{C}}^ m_ 0),\) and \(\mu_ A(f) \geq \nu (f).\) The author proves linear complementary inequalities in case I is prime. That is, if \({\mathcal O}\) is an integral domain, there are constants \(a_ 1,b_ 1,a_ 2,b_ 2,a_ 3\), and \(b_ 3\) so that \(\nu (fg) \leq a_ 1(\nu (f) + \nu (g)) + b_ 1,\) \(\mu_ A(f) \leq a_ 3\nu (f) + b_ 1,\) and \(\nu (f\circ \Phi) \leq a_ 2\nu (f) + b_ 2\) if the generic rank of \(\Phi\) is m.
Reviewer: G.Harris

MSC:

32B05 Analytic algebras and generalizations, preparation theorems
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14B05 Singularities in algebraic geometry
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