Generalisation of Polarization identity

[Geometry]

Hello,

If I have a quadratic form $q$ on a $\mathbb{R}$ vectorial space $E$, its associated bilinear symmetric form $b$ can be deduce by the following formula : $b(., .) = \frac{q(. + .) - q(.) - q(.)}{2}$. So that, an homogeneous polynomial of degree 2 can be associated to a blinear symmetric form.

Can we generalize this at a k multilinear symmetric form? (I mean associate to an homogeneous polynomial of degree k a k multilinear symmetric form).

Have a nice day.

 

[Geometry]

Their is a a way to developed the formal sum : $(x_1 + x_2 + ... + x_k)^n$ no? It might help but this will create a big sum with a lot of binomial coefficient.

 

[WWGD]

I think this is the/a generalization:

https://en.wikipedia.org/wiki/Polarization_of_an_algebraic_form

 

[Geometry]

Hello,

This is very clear and very adapted for my needs. Thank you very much WWGD.

I wish you a good day.