Extreme Polynomials and Multilinear Forms on l1

Given a 2-homogeneous polynomial P x, y.sax2qby2qcxy with real coeffi- cients, let 5P5rand 5P5cdenote the norms of P on the real and complex Banach space l12, respectively. We show 5P5rs5P5c, and obtain a sufficient and necessary condition on the coefficients a, b, and c for P to have norm 1. Applying these results, we characterize extreme points of the unit ball of P 2 l1. for the real Banach space l12 and examine them for the complex Banach space l12. We apply them to find extreme points and strongly extreme points of the unit ball of P 2 l1. and get an extremal 2-homogeneous polynomial on l1 that is not an extreme point. We also characterize extreme points and strongly extreme points of the unit ball of L ml1.or L mL1w0, 1x..