On the Noether and the Cayley-Bacharach theorems with PD multiplicities

In this paper, we prove the Noether theorem with the multiplicities described by PD operators. Despite the known analogue versions in this case the provided conditions are necessary and sufficient. We also prove the Cayley-Bacharach theorem with PD multiplicities. As far as we know this is the first generalization of this theorem in the case of multiple intersections.