Non-symmetric polarization

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Let P be an m-homogeneous polynomial in n -complex variables x₁,…,x_n

x_{1}, \dots, x_{n}

. Clearly, P has a unique representation in the form

P (x) = \sum_{1 \leq j_{1} \leq \dots \leq j_{m} \leq n} c_{(j_{1}, \dots, j_{m})} x_{j_{1}} \dots x_{j_{m}},

and the m-form

L_{P} (x^{(1)}, \dots, x^{(m)}) = \sum_{1 \leq j_{1} \leq \dots \leq j_{m} \leq n} c_{(j_{1}, \dots, j_{m})} x_{j_{1}}^{(1)} \dots x_{j_{m}}^{(m)}

satisfies e154179b08c98b0b41709c92" title="Click to view the MathML source">L_P(x,…,x)=P(x)

L_{P} (x, \dots, x) = P (x)

for every x∈Cⁿ

x \in C^{n}

. We show that, although L_P

L_{P}

in general is non-symmetric, for a large class of reasonable norms

‖ \cdot ‖

on Cⁿ

C^{n}

the norm of L_P

L_{P}

(C^{n}, ‖ \cdot {‖)}^{m}

up to a logarithmic term (clog⁡n)^m²

{(c \log n)}^{m^{2}}

can be estimated by the norm of P on

(C^{n}, ‖ \cdot ‖)

; here 30e6572fc7b46b1f8431e45a204f058" title="Click to view the MathML source">c≥1

c \geq 1

denotes a universal constant. Moreover, for the ℓ_p

ℓ_{p}

-norms

{‖ \cdot ‖}_{p}

, 1≤p<2

1 \leq p < 2

the logarithmic term in the number n of variables is even superfluous.