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 $\sqrt I$?



The Hilbert function of an intersection
It sure does help to assume complete intersection. It's nicer to describe the Hilbert series $H(t) := \sum\_k h(k) t^k$ than the Hilbert function $h(k)$. For the polynomial ring itself, it's $H(t) = 1/(1-t)^{n+1}$. For the complete intersection, it's $(1-t^d)^r/(1-t)^{n+1}$ (with an obvious generalization if the degrees are different).


If it's not a complete intersection, there's much less you can say -- you have to look at the syzygies of your generators, and the higher syzygies, and so on inclusion/exclusion.


Going to $\sqrt{I}$, all that's easy to say is that the degree of $h(k)$ stays the same but the leading coefficient can drop.


