Assume that $X_1,\ldots,X_r\subseteq\mathbb P^n$ are irreducible, reduced hypersurfaces in complex projective space, each of the same degree $d$. In other words, $X_i=Z_\ast(f_i)$ for certain irreducible, homogeneous polynomials $f_i\in\mathbb C[X_0,\ldots,X_n]_d$. Let $X:=X_1\cap\cdots\cap X_r$ be their intersection. For the moment, let's just say scheme-theoretic intersection. If it helps, assume that the codimension of $X$ is $r$, i.e. it is the complete intersection of the $X_i$. I would like to know what is known, if anything, about the Hilbert function of the closed subscheme $X$, or equivalently, the ideal $I=(f_1,\ldots,f_r)$. Can it be expressed in terms of the $f_i$ somehow?
Not very important, but still interesting: What about passing to $\sqrt I$?