Multiplicity and the Lojasiewicz exponent
نویسندگان
چکیده
منابع مشابه
On the Lojasiewicz Exponent of the Gradient of a Polynomial Function
Let h = ∑ hαβX Y β be a polynomial with complex coefficients. The Lojasiewicz exponent of the gradient of h at infinity is the upper bound of the set of all real λ such that |gradh(x, y)| ≥ c|(x, y)| in a neighbourhood of infinity in C, for c > 0. We estimate this quantity in terms of the Newton diagram of h. The equality is obtained in the nondegenerate case.
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The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative charact...
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Example 1. Set f1 = x d 1 and fi = xi−1 − x d i for i = 2, . . . , n. Then Φ(x) := maxi{|fi(x)|} > 0 for x 6= 0. Let p(t) = (t d , t n−2 , . . . , t). Then limt→0 ||p(t)||/|t| = 1 and Φ(p(t)) = t d . Thus the Lojasiewicz exponent is ≥ d. (In fact it equals d.) This works both over R and C. In the real case set F = ∑ f 2 i . Then degF = 2d, F has an isolated real zero at the origin and the Lojas...
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ژورنال
عنوان ژورنال: Banach Center Publications
سال: 1988
ISSN: 0137-6934,1730-6299
DOI: 10.4064/-20-1-353-364