Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is there a formula relating $mult_p(Z)$ and $mult_p(Z_T)$?
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The answer is yes (auniket's comment meant probably $T$ is generic and not $p$). This is a classical result : if $A$ is a noetherian local ring of dimension $d>0$, with infinite residue field, then there exists a system of parameters $(f_1,...,f_d)$ such that $\mathrm{mult}(A)=\mathrm{mult}(A/(f_1,...,f_{d1}))$, see ZariskiSamuel vol. II, Chap. VIII, §10, Theorem 22 and Remark page 296. Note that the result is false if the residue field is finite. But it is true that $\mathrm{mult}_p(Z)$ is a finite integral combination (with possibly negative coefficients) of multiplicities at $p$ of $Z\cap T_i$'s. See Proposition 5.9 in here. 

