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On the Łojasiewicz exponent of the gradient of a holomorphic function. (English) Zbl 0924.32007
Jakubczyk, Bronisław (ed.) et al., Singularities symposium – Łojasiewicz 70. Papers presented at the symposium on singularities on the occasion of the 70th birthday of Stanisław Łojasiewicz, Cracow, Poland, September 25–29, 1996 and the seminar on singularities and geometry, Warsaw, Poland, September 30–October 4, 1996. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 44, 149-166 (1998).
Author’s abstract: “The Łojasiewicz exponent of the gradient of a convergent power series \(h(X,Y)\) with complex coefficients is the greatest lower bound of the set of \(\lambda>0\) such that the inequality \(| \text{grad} (x,y)|\geq c|(x,y)|^\lambda\) holds near \(0\in\mathbb{C}^2\) for a certain \(c>0\).
We give an estimate of the Łojasiewicz exponent of \(\text{grad} h\) using information from the Newton diagram of \(h\). We obtain the exact value of the exponent for non-degenerate series”.
For the entire collection see [Zbl 0906.00013].

MSC:
32B15 Analytic subsets of affine space
32C25 Analytic subsets and submanifolds
32S05 Local complex singularities
32B99 Local analytic geometry
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