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Examples of Artin functions of germs of analytic spaces. (Exemples de fonctions de Artin de germes d’espaces analytiques.) (French) Zbl 1135.32024
Brasselet, Jean-Paul (ed.) et al., Singularities in geometry and topology 2004. Proceedings of the 3rd Franco-Japanese colloquium on singularities, Hokkaido, Japan, September 13–18, 2004. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-39-6/hbk). Advanced Studies in Pure Mathematics 46, 197-210 (2007).
This is a mainly expository article about Artin functions. An $$A$$-function for a set of series $$f_1,\dots, f_r$$ in $$k[[T, X]]:= k[[T_1,\dots, T_N, X_1,\dots, X_n]]$$ ($$k$$ a field) is a mapping $$\beta:\mathbb{N}\to\mathbb{N}$$ such that, for all $$p$$, if $$(x_1(T),\dots, x_n(T))= x(T)\in x(T)\in M^n$$ (with $$M$$ the maximal ideal of $$k[[T]]$$) such that $$f_i(T)\in M^{\beta(p)+ 1}$$ for all $$i$$, then there is an $$n$$-tuple $$\overline x(T)= (\overline x_1(T),\dots,\overline x_n(T),\overline x_i\in k[[T]]$$ for all $$i$$, such that $$f_i(T,\overline x(T))= 0$$ and $$x_i(T)-\overline x_i(T)\in M^{p+1}$$, for all $$i$$. If there are $$A$$-functions, then there is a smallest one, called the Artin function of $$f_1,\dots,f_r$$. This depends on the ideal of $$k[[T, X]]$$ spanned by $$f_1,\dots, f_r$$ only. If $$k$$ is a complete valued field of characteristic zero, then the Artin function of $$f_1,\dots, f_r$$ as above are always defined.
If $$g_1,\dots, g_r$$ are elements of $$k\{X_1,\dots, X_n\}$$ (convergent power series, $$k$$ a complete valued field), defining a germ of analytic space $$(Y, 0)\subseteq (k^n, 0)$$, the $$N$$th Artin function $$\beta_N$$ $$(N= 1,2,\dots)$$ of this germ is the Artin function of the ideal that $$g_1,\dots, g_r$$ span in $$k[[T_1,\dots, T_N, X_1,\dots, X_n]]$$. More precisely, this paper deals with the Artin functions of a germ as above. Firstly, there is a discussion of basic properties of Artin functions. For instance, the germ is nonsingular if and only if an Artin function $$\beta_N$$ thereof is not the identity (thus this functions are a sort of measure of the singularity). Next there are results about inequalities and bounds satisfied (under suitable assunptions) by these functions. Aside from other more elementary results, on the basis of a “Diophantine approximation” theorem (proved elsewhere by the author), one obtains that the Artin function of a homogeneous polynomial $$P(X_1, X_2)$$ with coefficients in $$k[[T_1,\dots,T_N]]$$ is bounded by an affine function $$h(p)=(d+ a)p+ c$$, where $$d$$ is the degree of $$P$$, and $$a\geq 1$$, $$c$$ are constants. The last section is devoted to examples. Explicit calculations of $$\beta_N$$ (any $$N$$) are given for a germ $$(Y, 0)\subset(k^n, 0)$$ defined by a monomial or when all the irreducible components are smooth. For $$\beta_1$$ (also called the Greenberg function) calculations are given for a germ of hypersurface defined by a monomial, or by the equation $$X_1X_2- X_3X_4$$ $$(n= 4)$$ and also for a cusp $$(N= 2)$$. For general $$N$$ interesting inequalities are obtained for the cusp and the Whitney umbrella.
There is an extensive bibliography. The paper is well written and is a good introduction to this difficult and interesting subject.
For the entire collection see [Zbl 1111.58001].
##### MSC:
 32S20 Global theory of complex singularities; cohomological properties 32S05 Local complex singularities 14B12 Local deformation theory, Artin approximation, etc. 32S10 Invariants of analytic local rings 32S99 Complex singularities