Linear Chevalley estimates

Author:
Ti Wang

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4877-4898

MSC:
Primary 32S05; Secondary 32B10, 32S10

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308027-5

MathSciNet review:
1308027

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Abstract | References | Similar Articles | Additional Information

Abstract: A *Chevalley estimate* for a germ of an analytic mapping is a function such that if the composite with of a germ of an analytic function on the target vanishes to order at least , then it vanishes on the image to order at least . Work of Izumi revealed the equivalence between regularity of a mapping (in the sense of Gabrielov, see ) and the existence of a linear Chevalley estimate . Bierstone and Milman showed that uniformity of the Chevalley estimate is fundamental to several analytic and geometric problems on the images of mappings. The central topic of this article is uniformity of linear Chevalley estimates for regular mappings.

We first establish the equivalence between uniformity of a linear Chevalley estimate and uniformity of a "linear product estimate" on the image: A *linear product estimate* on a local analytic ring (or, equivalently, on a germ of an analytic space) means a bound on the order of vanishing of a product of elements which is linear with respect to the sum of the orders of its factors. We study the linear product estimate in the central case of a hypersurface (i.e., the zero set of an analytic function). Our results show that a linear product estimate is equivalent to an explicit estimate concerning resultants. In the special case of hypersurfaces of multiplicity , this allows us to prove uniformity of linear product estimates.

**[BM1]**E. Bierstone and P. Milman,*The local geometry of analytic mappings*, Dottorato di ricerca in matematica Universita di Pisa, Dipartimento di Matematica, 1988. MR**971251 (90j:32011)****[BM2]**-,*Relations among analytic functions*, Ann. Inst. Fourier (Grenoble)**37:1, 37:2**(1987).**[BM3]**-,*The Newton diagram of an analytic morphism, and applications to differentiable functions*, Bull. Amer. Math. Soc.**9**(1983). MR**714993 (85g:58016)****[BM4]**-,*Geometric and differential properties of subanalytic sets*, Bull. Amer. Math. Soc.**25**(1991). MR**1102751 (92h:32008)****[BM5]**-,*Arc-analytic functions*, Invent. Math.**101**(1990), 411-424. MR**1062969 (92a:32011)****[BM6]**-,*A simple constructive proof of canonical resolution of singularities, effective methods in algebraic geometry*, Progress in Math.**94**(1991), 11-30. MR**1106412 (92h:32053)****[EH]**P.M. Eakin and G.A. Harris,*When**convergent implies**is convergent*, Math. Ann.**229**(1977), 201-210. MR**0444651 (56:3001)****[Ga]**A. M. Gabrielov,*Formal relations between analytic functions*, Math. USSR Izv.**7**(1973), 1056-1088. MR**0346184 (49:10910)****[G1]**G. Glaeser,*Fonctions composées différentiales*, Ann. of Math.**77**(1963), 193-209. MR**0143058 (26:624)****[GR]**H. Grauert and R. Remmert,*Coherent analytic sheaves*, Springer-Verlag, 1984. MR**755331 (86a:32001)****[I1]**S. Izumi,*Linear complementary inequalities for orders of germs of analytic functions*, Invent. Math.**65**(1982), 459-471. MR**643564 (83k:32014)****[I2]**-,*A measure of integrity for local analytic algebras*, Publ. RIMS Kyoto Univ.**21**(1985), 719-735. MR**817161 (87i:32014)****[I3]**-,*Gabrielov's rank condition is equivalent to an inequality of reduced orders*, Math. Ann.**276**(1986), 81-89. MR**863708 (87j:32021)****[Ma]**H. Matsumura,*Commutative algebra*, Benjamin/Cummings, Reading, Mass., 1980. MR**575344 (82i:13003)****[Mi]**P. Milman,*Analytic and polynomial homomorphisms of analytic rings*, Math. Ann.**232**(1978), 247-253. MR**0492359 (58:11486)****[Na]**M. Nagata,*Local rings*, Interscience, 1962. MR**0155856 (27:5790)****[R1]**D. Rees,*Izumi's Theorem, commutative algebra*, Springer-Verlag, 1989. MR**1015531 (90g:13010)****[R2]**-,*Valuations associated with a local ring*. I, Proc. London Math. Soc.**5**(1955), 107-128. MR**0067095 (16:669c)****[R3]**-,*Valuations associated with a local ring*. II, Proc. London Math. Soc.**31**(1956), 228-235. MR**0078972 (18:8c)****[T1]**J-Cl. Tougeron,*Idéaux de fonctions différentiables*, Berlin-Heidelberg-New York, 1972.**[T2]**-,*Sur les racines d'un polynome a coefficients series formells, real analytic and algebraic geometry*, Lecture Notes in Math., vol. 1420, Springer-Verlag, 1990.**[W1]**T. Wang,*A stratification given by Artin-Rees estimates*, Canad. J. Math.**44:1**(1992). MR**1152675 (93e:13031)****[W2]**-,*Linear inequalities in local analytic geometry*, Doctoral Dissertation, University of Toronto, 1992.**[Wh]**H. Whitney,*Complex analytic varieties*, Addison-Wesley, 1972. MR**0387634 (52:8473)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308027-5

Keywords:
Chevalley estimates,
Izumi's Theorem,
irreducibility

Article copyright:
© Copyright 1995
American Mathematical Society