  We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size
$v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such
that every $t$-element subset of $X$ is contained in exactly $\lambda$ "blocks"
$B\in \mathcal{B}$ for some fixed $\lambda$. We represent each block $B$ by a
zero-one vector $\mathbf{c}_B$ of length $v$ and explore the ideal
$\mathcal{I}(\mathcal{B})$ of polynomials in $v$ variables with complex
coefficients which vanish on the set $\{ \mathbf{c}_B \mid B \in
\mathcal{B}\}$. After setting up the basic theory, we investigate two
parameters related to this ideal: $\gamma_1(\mathcal{B})$ is the smallest
degree of a non-trivial polynomial in the ideal $\mathcal{I}(\mathcal{B})$ and
$\gamma_2(\mathcal{B})$ is the smallest integer $s$ such that
$\mathcal{I}(\mathcal{B})$ is generated by a set of polynomials of degree at
most $s$. We first prove the general bounds $t/2 < \gamma_1(\mathcal{B}) \le
\gamma_2(\mathcal{B}) \le k$. Examining important families of examples, we find
that, for symmetric 2-designs and Steiner systems, we have
$\gamma_2(\mathcal{B}) \le t$. But we expect $\gamma_2(\mathcal{B})$ to be
closer to $k$ for less structured designs and we indicate this by constructing
infinitely many triple systems satisfying $\gamma_2(\mathcal{B})=k$.
