Vector invariants in arbitrary characteristic

Let k be an algebraically closed field of characteristic p > 0. Let H be a subgroup of GL(n,k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d > n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always integral over the algebra of polarized invariants and when H is reductive is actually the p - root closure of that algebra. We also give conditions for the algebras to be equal, relating equality to good filtrations and saturated subgroups. We conclude with examples where H is finite or a classical group or is a certain kind of unipotent subgroup of GL(n,k).