Non-symmetric polarization

Let P be an m-homogeneous polynomial in n  -complex variables x1,…,xnx1,…,xn. Clearly, P has a unique representation in the formP(x)=∑1≤j1≤…≤jm≤nc(j1,…,jm)xj1⋯xjm, and the m-formLP(x(1),…,x(m))=∑1≤j1≤…≤jm≤nc(j1,…,jm)xj1(1)⋯xjm(m) satisfies LP(x,…,x)=P(x)LP(x,…,x)=P(x) for every x∈Cnx∈Cn. We show that, although LPLP in general is non-symmetric, for a large class of reasonable norms ‖⋅‖ on CnCn the norm of LPLP on (Cn,‖⋅‖)m up to a logarithmic term (clog⁡n)m2(clog⁡n)m2 can be estimated by the norm of P   on (Cn,‖⋅‖); here c≥1c≥1 denotes a universal constant. Moreover, for the ℓpℓp-norms ‖⋅‖p, 1≤p<21≤p<2 the logarithmic term in the number n of variables is even superfluous.