There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be the local normalization of $V_f$ then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t)))$. The other one is given in Algebraic Geometry by Hartshorne. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Do these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1))+\mathrm{length}(O_P/(g, f_2))$ is also important.

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    $\begingroup$ If your algebraic surface is smooth, a good (conceptual) way to compare the two is by comparing both to the cup product in cohomology. $\endgroup$ Jun 1 '20 at 3:48
  • $\begingroup$ A proof (for the $n$-dimensional version) is given in Chapter 4 of arxiv.org/abs/1806.05346. Sorry for the self-promotion, but one reason for including this element in the (draft of the) book was that I could not locate a direct proof in the literature. $\endgroup$
    – auniket
    Jun 1 '20 at 20:19

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