Finite field Nullstellensatz and Grassmannians

Let X C ]p'N be a projective variety defined over the Galois field GF(q). Denote by X(q) the set of GF(q)-rational points of X. Let k be an integer. We say that the pair (X, X(q)) satisfies the Finite Field Nullstellensatz of order k, (the FFN(k), for short), if every homogeneous form of degree :::; k on r N (J<) vanishing on X (q), vanishes on X (J<). Here, we prove the Finite Field Nullstellensatz FFN(q) for any Grassmann variety.