## 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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### References:

 [1] Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. IHES, 36 (1969), 23-58. · Zbl 0181.48802 [2] Becker, J., On the composition of power series, In : Commutative algebra (analytic methods (LN in pure and ap. math. 68), (159-172), Marcel Dekker New York, 1982. · Zbl 0514.13015 [3] Brieskorn, E., Uber die Auflosung gewisser Singularitaten von holomorphen Abbild- ungen, Math. Ann., 166 (1966), 76-102. · Zbl 0145.09402 [4] Flenner, EL, Die Satze von Bertini fur lokale Ringe, Math. Ann., 229 (1977), 97-111. · Zbl 0398.13013 [5] Grauert, H., Uber Modifikationen und exceptionelle analytische Mengen, Math. Ann., 146 (1962), 331-368. · Zbl 0173.33004 [6] Grauert, H. and Remmert, R., Analytische Stellenalgebren (GMW 176), Springer, Berlin-Heidelberg-New York, 1971. [7] Galbiati, M., Tognoli, A., Alcune proprieta delle varieta algebriche reali, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 359-404. · Zbl 0277.14010 [8] Hironaka, H., Subanalytic sets, In : Number theory, algebraic geometry and commutative algebra, in honor of Y. Akizuki, (453-493), Kinokuniya, Tokyo, 1973. [9] Izumi, S.j Linear complementary inequalities for orders of germs of analytic functions, Invent. Math., 65 (1982), 459-471. · Zbl 0497.32005 [10] , Inequalities for orders on a rational singularity of a surface, J. Math. Kyoto Univ., 24 (1984), 239-241. · Zbl 0561.14017 [11] , Lejeune-Jalabert, M. and Teissier, B., Cloture integrate des ideaux et equisin- gularite, ficole Polytechnique 1974. [12] Malgrange, B., Sur les fonctions differentiates et les ensenbles analytiques, Bull. Soc. Math. France, 91 (1963), 113-127. · Zbl 0113.06302 [13] Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Pub. Math. IHES, II (1961), 229-246. · Zbl 0108.16801 [14] Nagata, M., Note on a paper of Samuel concerning asymptotic properties of ideals, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 30 (1957), 165-175. · Zbl 0095.02305 [15] , Local rings (Interscience tracts in Pure & applied Math. 13), Interscience Publishers, New York-London, 1962. [16] Rees, D., Valuations associated with a local ring (I), Proc. London Math. Soc., 5(1955), 107-128. · Zbl 0066.28805 [17] , Valuations associated with ideals (II), /. London Math. Soc., 31 (1956), 221-228. · Zbl 0074.26303 [18] , Valuations associated with a local ring (II), /. London Math. Soc., 31 (1956), 228-235. · Zbl 0074.26401 [19] Risler, J-J., Le theorem des zeros en geometries algebraique et analytique reelles, Bull. Soc. Math. France, 104 (1976), 113-127. · Zbl 0328.14001 [20] Samuel, P., Some asymptotic properties of powers of ideals, Ann. of Math., 56 (1952), 11-21. · Zbl 0049.02301
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