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Multiplicity and the Łojasiewicz exponent. (English) Zbl 0661.32018
Singularities, Banach Cent. Publ. 20, 353-364 (1988).
[For the entire collection see Zbl 0653.00009.]
The present paper contains a survey of some recent results concerning the Lojasiewicz exponent of a holomorphic mapping at an isolated zero. In some cases, the author presents alternative proofs of earlier results. A comparison of the two invariants of a holomorphic mapping f - its multiplicity \(m_ 0(f)\) and Lojasiewicz exponent \(\ell_ 0(f)\)- has been made and some further properties of Lojasiewicz exponent have also been obtained. For instance, the author gives a new proof of his following earlier result [Bull. Pol. Acad. Sci., Math. 32, 669-673 (1984; Zbl 0579.32043)]: If \(f=(f_ 1,...,f_ n): ({\mathbb{C}}^ n,0)\to ({\mathbb{C}}^ n,0)\) be a finite holomorphic mapping, then \[ \max^{n}_{i=1}(ord f_ i)\leq \ell_ 0(f)\leq m_ 0(f)- \prod^{n}_{i=1}ord f_ i+ \max^{n}_{i=1}(ord f_ i). \] As a corollary, it is deduced that if ord \(f_ 1=...=ord f_ n=k\) and \(m_ 0(f)=k^ n+1,\) then \(\ell_ 0(f)=k+1\).
Reviewer: O.P.Juneja

32B99 Local analytic geometry