Computation of the Łojasiewicz exponent of nonnegative and nondegenerate analytic functions
Computation of the Łojasiewicz exponent of nonnegative and nondegenerate analytic functions
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非负和非简并解析函数的 Łojasiewicz 指数的计算
DOI：
10.1142/s0129167x1450092x
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发表时间：
2014
影响因子：
0.6
通讯作者：
T. Pham
中科院分区：
文献类型：
--
作者：
Nguyễn Thao Nguyên Búi;T. Pham
设f：(ℝn，0)→(ℝ，0)是定义在原点0∈ℝn的邻域内的非常数解析函数.经典的Łojasiewicz不等式指出，存在正常数δ，c和L，使得|f(X)|≥CD(x，f-1(0))L对于‖x‖≤δ，其中d(x，f-1(0))表示从x到集合f-1(0)的距离.F的Łojasiewicz指数在原点0∈ℝn处，记为$\mathscr{L}0(F)$，是满足Łojasiewicz不等式的指数L的下确界。本文在f为非负、非退化的情况下，利用f的牛顿多面体建立了f的Łojasiewicz指数$\mathscr{L}_0(F)$的计算公式。
Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for ‖x‖ ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by $\mathscr{L}_0 (f)$, is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent $\mathscr{L}_0 (f)$ of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate.