Two Problems of the Theory of Quadratic Maps

Given quadratic forms $q_1, \dots, q_k$, two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point $x=0$? When is the maximum of these quadratic forms nonnegative or positive for any $x\ne 0$? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.