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Let p be a prime and K the algebraic closure of the finite field GFepT. We will always work in characteristic p and consider P, as a scheme over GFepT. Let X be an alge-braic scheme defined over a finite field GFep,T. X eKT will denote the set of all K-points of X. For every power q of p with q d p, let X eqT denote the set of all GFeqT-points of X. Hence X eqT J X eq,T if q, q, are p-powers and q, dqdp,. X eKT is the union of all X eqT, q g 0 and q a p-power. If X is reduced, then the scheme X is uniquely deter-mined by the algebraic variety X eKT in the sense of Serre (Hilbert Nullstellensatz). If X is not a zero-dimensional scheme, then X eKT is infinite. We fix a p-power q with q d p, and we would like to see up to what order the finite set X eqT determines the infinite set X eKT. Now assume that X is projective and that it is equipped with an embedding XHP, defined over GFeqT. Let k be an integer. We say that the pair eX ; X eqTT satisfies the Finite Field Nullstellensatz of order k (or just that FFNekT is true for X and X eqT) if every homogeneous form of degree c k on P,eKT vanishing on X eqT vanishes on X eKT. Choose homogeneous coordinates z0;... ; zN on P,. The set PGeN; qT is the union of q t 1 hyperplanes; for instance take the hyperplanes z0 ? czN, c A GFeqT, and the hyperplane zN ? 0.Hence if X eKT 0 X eqT (and in particular if dimeX T > 0), then the pair eX ; X eqTT does not satisfy FFNeq t 1T. A. Blokhuis and G. E. Moor-house proved FFNeq 1T for an elliptic quadric surface, FFNeqT for a hyperbolic quadric surface and FFNeqT for a smooth quadric hypersurface of PGen; qT, q d 4 [1]. G. E. Moorhouse proved FFNeqT for Hermitian varieties, q a square [5, Theorem 4.1], and FFNeq 1T for Grassmann varieties [6, O4]. Here we consider the case of the intersection of two quadric hypersurfaces and prove the following result.

On the Finite Field Nullstellensatz for the intersection of two quadric hypersurfaces