Let \(\mathbb{P}^n\) be the \(n\)-dimensional projective space defined over an algebraically closed field of positive characteristic \(p\), let \(q\) be a power of \(p\) and let \(X\) be an irreducible closed \(\mathbb{F}_q\)-definable subvariety of \(\mathbb{P}^n\) (here \(\mathbb{F}_q\) denotes the Galois field of \(q\) elements). The authors introduce the following terminology: For a given natural number \(k\) one says that the projective variety \(X\) has the Finite Field Nullstellensatz property \(\text{FFN}(k,q)\) if any \(n+1\)-variate form of degree \(k\) over \(\mathbb{F}_q\) which vanishes on the \(\mathbb{F}_q\)-rational points of \(X\), vanishes also on all points of \(X\). The authors show that any \(d\)-fold Veronese variety has the property \(\text{FFN}(k,q)\) if and only if the numerical condition \(dk\leq q\) is satisfied. Moreover they prove that the Segre variety has the property \(\text{FFN}(q,q)\). The main result of the paper is the following: Suppose that \(X\) is of dimension \(m