On the multiplicity of holomorphic maps and a residue formula (Q1313440)

The (algebraic or geometric) multiplicity of holomorphic map \((f_ 1, \dots, f_ n)\) at an isolated zero \(0 \in \mathbb{C}^ n\) can be expressed as the local residue at 0 of \({df_ 1 \over f_ 1} \wedge \cdots \wedge {df_ n \over f_ n}\), which by means of the Bochner-Martinelli kernel can be written as an integral over a small \((2n-1)\)-ball around the origin. Under appropriate hypotheses and based on Stokes theorem, the author presents integral formulas over \((2n-2p-1)\) dimensional cycles (for \(1 \leq p \leq n-1)\) for the computation of this local multiplicity.