Generalisation of Polarization identity

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SUMMARY

The discussion centers on the generalization of the polarization identity for quadratic forms to k multilinear symmetric forms. It establishes that a bilinear symmetric form can be derived from a quadratic form using the formula: b(., .) = (q(. + .) - q(.) - q(.))/2. The participants explore the possibility of associating an homogeneous polynomial of degree k with a k multilinear symmetric form, referencing the polarization of algebraic forms as a foundational concept.

PREREQUISITES
  • Understanding of quadratic forms and their properties
  • Familiarity with bilinear symmetric forms
  • Knowledge of homogeneous polynomials and their degrees
  • Basic concepts of multilinear algebra
NEXT STEPS
  • Research the generalization of the polarization identity for multilinear forms
  • Study the properties of homogeneous polynomials of degree k
  • Explore the application of binomial coefficients in polynomial expansions
  • Examine the Wikipedia page on the polarization of algebraic forms for further insights
USEFUL FOR

Mathematicians, students of algebra, and researchers interested in multilinear algebra and the properties of polynomial forms will benefit from this discussion.

Geometry
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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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Likes   Reactions: WWGD

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