BRIEF COMMUNICATIONS Two Problems of the Theory of Quadratic Maps

Given quadratic forms q1 ,...,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 �= 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. q1(x )=( x 1 ) 2 − (x 2 ) 2 , q2(x )=2 x 1 x 2 ,a ndq3(x )= −(x 1 ) 2 − (x 2 ) 2 +( x 3 ) 2 + ··· +( x n ) 2 , but there exists no y with the properties specified above in this case.