Before revision: We show that^{\lambda_i} be finite sets, and let S:=S_1 \times S_2 \times \ldots \times S_m \subset C^n be the multi-grid defined by S_i. Suppose p is an n-variate degree d polynomial.
After revision: Motivated by applications in combinatorial geometry, we consider the following question: Let \lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m) be an m-partition of a positive integer n, S_i \subseteq \mathbb{C^{\lambda_i} be finite sets, and let S:=S_1 \times S_2 \times \ldots \times S_m \subset C^n be the multi-grid defined by S_i. Suppose p is an n-variate degree d polynomial.