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On the order of holomorphic and \(c\)-holomorphic functions. (English) Zbl 1176.32002

Summary: In the first part of this paper, we prove that the Łojasiewicz exponent of a non-constant holomorphic germ \(f:(\mathbb C^m,0)\to (\mathbb C,0)\) is a good exponent for \(f\) coinciding with the order of vanishing of \(f\) at zero and the degree at zero of its cycle of zeros \(Z_f\). As an application of this result, we show that for any holomorphic curve germ \(\gamma: (\mathbb C,0)\to (\mathbb C^m,0)\) one has \(\text{ord}_0(f\circ \gamma) = \text{ord}_0f\cdot \text{ord}_0\gamma\) if and only if is transversal to \(f^{-1}(0)\) at zero.
In a recent paper, we have introduced an order of flatness for c-holomorphic functions which allowed us to give some bounds on the Łojasiewicz exponent of c-holomorphic mappings. Answering a question of A. Płoski, we show that both notions (the order of flatness and the Łojasiewicz exponent) are intrinsic to the analytic set given (this allows to carry these notions over to analytic spaces). We then turn to considerations about possible Łojasiewicz exponents of c-holomorphic mappings.
The last part deals with quotients of c-holomorphic functions. We investigate relations between this newly introduced order of flatness and the possibility of dividing one c-holomorphic function by another.

MSC:

32A10 Holomorphic functions of several complex variables
32A17 Special families of functions of several complex variables
32C20 Normal analytic spaces
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