Extreme Polynomials and Multilinear Forms onl₁

Given a 2-homogeneous polynomialP(x,y)=ax²+by²+cxywith real coefficients, let P_rand P_cdenote the norms ofPon the real and complex Banach spacel²₁, respectively. We show P_r=P_c, and obtain a sufficient and necessary condition on the coefficientsa,b, andcforPto have norm 1. Applying these results, we characterize extreme points of the unit ball of P(²l₁) for the real Banach spacel²₁and examine them for the complex Banach spacel²₁. We apply them to find extreme points and strongly extreme points of the unit ball of P(²l₁) and get an extremal 2-homogeneous polynomial onl₁that is not an extreme point. We also characterize extreme points and strongly extreme points of the unit ball of L(^ml₁) or L(^mL₁[0,1]).