Abstract
We discuss a formula of S. Spodzieja and generalize it for the isolated improper Achilles-Tworzewski-Winiarski intersection index. As an application we give a simple proof of a result of P. Ebenfelt and L. Rothschild: if $${F\colon (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)}$$ is a finite holomorphic map, W a germ of a complex variety at zero such that F −1(W) is a smooth germ and the Jacobian of F does not vanish identically on it, then W is smooth too.
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