Norm of $n$-linear symmetric forms

After a bit of searching, I found that $\gamma_n=1$ for all $n$.

This is known as "van der Corput-Schaake inequality" (1935), discovered before by Szegö (1928), and mentioned to have been known to Banach in the Scottish Book.

The proof proceeds by showing that if $P$ is homogeneous of degree $n$ on a euclidean space, one has, for $B$ the unit ball, $$\sup_{B\times B} |\nabla P(x)\cdot y|\leq n \sup_B |P|,$$ reducing first to the 2-dimensional case and reasoning about trigonometric polynomials.

This implies by differentiating successively that $$\sup_{x_i\in B}|D^nP(x_1,\dots,x_n)| \leq n! \sup_B |P|,$$ hence the result since $D^nP/n!$ is the polarization of $P$.

I must admit that I don't understand the proof yet.

Here is a link to van der Corput-Schaake's article

http://archive.numdam.org/article/CM_1935__2__321_0.pdf

EDIT: I found more recent references with simpler proofs, for instance in Chapter 4 of DeVore and Lorentz "Constructive approximation" (Springer 1993), cf the Szegö inequality, and in an expository text by Lawrence A. Harris. Hope this helps.

The question is developed here :

http://www.math.tsukuba.ac.jp/~wkbysh/note3.pdf