Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by $$ I_X =I_1 \cap \ldots \cap I_2 $$ I was wondering if the Hilbert polynomial of $p_{X}(t)$ can be given in terms of the Hilbert polynomials associated to the schemes defined by $I_i$ and their intersections. For example, adding a zero dimensional scheme $l_p \in \mathbb{P}^r$ increases the Hilbert polynomial by $h^0(\mathcal{O}_{l_p})$. Similarly, adding a component, $Y$, of dimension equal to $\max(dim(\mathcal{Z}(I_i)))$ increases the highest degree's coefficient.

I collected examples here and there ...but I am wondering if there is a big picture that I am missing.

Thanks!