Some remarks on non-symmetric polarization

Let P:Cn→C be an m-homogeneous polynomial given byP(x)=∑1≤j1≤…≤jm≤ncj1…jmxj1…xjm. Defant and Schlüters defined a non-symmetric associated m-form LP:(Cn)m→C byLP(x(1),…,x(m))=∑1≤j1≤…≤jm≤ncj1…jmxj1(1)…xjm(m). They estimated the norm of LP on (Cn,‖⋅‖)m by the norm of P on (Cn,‖⋅‖) times a (clog⁡n)m2 factor for every 1-unconditional norm ‖⋅‖ on Cn. A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schlüters' argument) brings the constant term down to (cmlog⁡n)m−1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than (clog⁡n)m/2 when n≫m. Finally, the case of ℓp-norms ‖⋅‖p with 1≤p<2 is addressed.