On an Oppenheim-type conjecture for systems of quadratic forms

Let Q(i), i = 1, . . . , t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set {(Q(1) ((x) over bar), . . . , Q(t)((x) over bar)): x is an element of Z(d) - {(0) over bar}} contains (0 , . . . , 0). As a corollary, we deduce several results on the magnitude of the set Delta of g is an element of GL(d, R.) such that the closure of the set {(Q(1)((x) over bar), . . . , Q(t)(g (x) over bar)) : (x) over bar is an element of Z(d) - {(0) over bar}} contains (0, . . . , 0). Special cases are described when, depending on the mutual position of the hypersurfaces {Q(i) = 0}, i = 1, . . . , t, the set Delta has full Haar measure or measure zero and Hausdorff dimension d(2) - (d - 2)/2.