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Lojasiewicz type inequalities and Newton diagrams. (English) Zbl 0737.58001
A holomorphic function defined near the origin $$0\in \mathbb{C}^ n$$ has an isolated singularity at 0 iff there are positive numbers $$\alpha$$, $$K$$ such that the following Lojasiewicz type inequality (L) holds near $$0: |\hbox{grad }f(x)|\geqq K| x|^ \alpha$$. Let $$\alpha_ 0(f)$$ be the minimal number of $$\alpha$$ such that $$(L_ \alpha)$$ holds near 0. Let $$\sum a_ nx^ n$$ be the Taylor expansion of $$f$$ at 0. Set $$\Gamma_ +(f)=\hbox{the convex hull of }\{\nu+R+^ n\mid a_ n\neq 0\}$$, $$l(\alpha):=\min\{\langle a,\alpha\rangle\mid a\in\Gamma_ +(f)\}$$, $$\gamma(\alpha):=\{a\in\Gamma_ +(f)\mid \langle a,\alpha\rangle =1(\alpha)\}$$. A vector $$a$$ supports a face $$\gamma$$ of $$\Gamma_ +(f)$$ if $$\gamma=\gamma(a)$$. $$f$$ is nondegenerate if the equations $$\partial f/\partial x_ 1=0,\dots,\partial f/\partial x_ n=0$$ have no common solutions on $$x_ 1,\dots,x_ n\neq 0$$ for any compact face $$\gamma\in\Gamma_ +(f)$$. Let $$V(f)$$ be the set of primitive vectors $$a$$ such that $$\gamma(a)$$ is an $$(n-1)$$-dimensional face of $$\Gamma_ +(f)$$. For a subset $$A$$ of $$V(f)$$ is set $$\varphi(A):=\max_{c_ i\geq 0} \min_ j\{\sum_ i c_ ia_ j^ i/\sum_ i c_ il(a^ i)\}$$ where $$A=\{a_ 1,\dots,a_ n\}$$. The rational number $$m_ 0(f)$$ is defined by the formula $m_ 0(f)=\max_{A\subset V(f)}\{\varphi^{-1}(A)\mid \sum_{a\in A}R_ +a\hbox{ is a cone of }\Gamma^*(f)\hbox{ and }\varphi(A)\neq 0\}.$ The main result of the paper is the Theorem. Let $$f$$ be an analytic function near 0, and $$f$$ has an isolated singularity at 0 and is nondegenerate in the sense of the above definition. Then $$\alpha_ 0(f)\leq m_ 0(f)-1$$.

##### MSC:
 58A20 Jets in global analysis 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
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