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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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