Is there a version of Bézout's theorem for non-proper intersections?

For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a scheme) $\bigcap_{j=1}^4 \mathbf{Z}(P_j)$ is a 1-dimensional variety in $\mathbb{C}^4,$ plus some embedded points. I would like to say that the number of embedded points is at most $\prod_{j=1}^4 \operatorname{deg}(P_j)$. Does anyone have a reference / counter-example for such a statement?

I would be happy to lose a constant as well; a bound of $O\Big(\prod_{j=1}^4 \operatorname{deg}(P_j)\Big)$ would be fine too.