Non-symmetric polarization

Let P be an m-homogeneous polynomial in n-complex variables x(1),..., x(n). Clearly, P has a unique representation in the form P(x) = Sigma(1 <= j1 <=...<= jm <= n) c(j(1),...,j(m)) x(j1)...x(jm), and the m-form L-P(x((1)),...,x((m)))= Sigma(1 <= j1 <=...<= jm <= n) c(j(1),...,j(m)) x(j1)((1))...x(jm)((m)), satisfies L-P(x,...,x) = P(x) for every x is an element of C-n. We show that, although L-P in general is non-symmetric, for a large class of reasonable norms parallel to center dot parallel to on C-n the norm of L-P on (C-n, parallel to center dot parallel to)(m) up to a logarithmic term (c log n)(m2) can be estimated by the norm of P on (C-n parallel to center dot parallel to); here c >= 1 denotes a universal constant. Moreover, for the l(p)-norms parallel to center dot parallel to(p), 1 <= p < 2 the logarithmic term in the number n of variables is even superfluous. (C) 2016 Published by Elsevier Inc.