For S\subseteq \mathbb{F}^n, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted \mathrm{h}_S(d,\mathbb{F}), is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets S of arbitrary finite grids in \mathbb{F}^n ... more >>>