Generalisation of Polarization identity

  • Context: Undergrad 
  • Thread starter Thread starter Geometry
  • Start date Start date
  • Tags Tags
    Identity Polarization
Click For Summary

Discussion Overview

The discussion revolves around the generalization of the polarization identity from quadratic forms to k multilinear symmetric forms. Participants explore the relationship between homogeneous polynomials of degree k and their associated multilinear symmetric forms.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant presents a formula for deriving a bilinear symmetric form from a quadratic form and questions whether this can be generalized to k multilinear symmetric forms.
  • Another participant suggests that developing the formal sum of powers might provide insight, although it would result in a complex expression involving binomial coefficients.
  • A third participant references a Wikipedia article on the polarization of algebraic forms as a potential generalization.
  • A later reply expresses gratitude for the clarity of the information provided, indicating it meets their needs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the generalization process, and multiple approaches and references are presented without resolving the question.

Contextual Notes

The discussion does not clarify the assumptions underlying the proposed generalizations or the specific definitions of the terms used, leaving some aspects unresolved.

Geometry
Messages
6
Reaction score
2
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.
 
Physics news on Phys.org
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.
 
  • Like
Likes   Reactions: WWGD

Similar threads

Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
6K
Undergrad Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Generators of Galois group of ## X^n - \theta ##
  • · Replies 3 ·
Replies
3
Views
2K
Graduate The meaning of ##(ict, x_1,x_2,x_3)##
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
MHB Why is the characteristic different from 2?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Is the proof of these results correct?
  • · Replies 28 ·
Replies
28
Views
3K
Undergrad Canonical Form for quadratic equations *with* linear terms
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Splitting ring of polynomials - why is this result unfindable?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Clarifying a corollary about Quadratic Forms
  • · Replies 6 ·
Replies
6
Views
2K