Undergrad Generalisation of Polarization identity

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A quadratic form on a real vector space can be associated with a bilinear symmetric form using the formula b(., .) = (q(. + .) - q(.) - q(.))/2. The discussion explores the possibility of generalizing this relationship to k multilinear symmetric forms, linking them to homogeneous polynomials of degree k. The user mentions the formal sum (x_1 + x_2 + ... + x_k)^n as a potential method for this generalization, despite the complexity introduced by binomial coefficients. A reference to the Wikipedia page on the polarization of algebraic forms is provided for further clarification. The conversation concludes with expressions of gratitude for the helpful insights.
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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.
 
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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.
 
Hello,

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

I wish you a good day.
 
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