##  [Polarization](/polarization-0) 

 Definition

A procedure that associates a symmetric multilinear form (the polarized form) to a homogeneous polynomial by evaluating the polynomial on linear combinations of variables and extracting multilinear coefficients; it 'linearizes' homogeneous polynomials.

 

 

 

 

 

 





## Principle

Principle

Use polarization identities: for a homogeneous polynomial p of degree n, define the n-linear form B(v1,...,vn) by an alternating sum or by ∂^n p at zero along linear directions; equivalently evaluate p on sums and divide by factorial combinations to recover multilinear symmetrized coefficients when factorials are invertible.

 

 

 

 

 





## Demonstration

Demonstration

A quadratic form q(x) polarizes to the symmetric bilinear form B(u,v) = (q(u+v) − q(u) − q(v) + q(0))/2. For homogeneous cubic polynomials similar finite difference formulas produce a symmetric trilinear form whose diagonal recovers the original cubic.

 

 

 

 

## Misapplication

Misapplication

Applying polarization without homogeneity or without handling characteristic issues (e.g., when n! is not invertible) can give incorrect or incomplete multilinearizations; treating polarization as a purely formal substitution may miss necessary normalization factors.

 

 

 

 

 





## Consequence

Consequence

Polarization produces symmetric multilinear forms that encode the coefficients of the polynomial and permit reconstruction of the polynomial by evaluating the multilinear form on diagonal inputs; it links polynomial invariants to multilinear and representation-theoretic objects.

 

 

 

 

## Reversal

Reversal

The diagonal (evaluation of the multilinear form at repeated arguments) is the reverse operation: given a symmetric n-linear form B, the homogeneous polynomial p(x) = B(x,...,x) recovers the original polynomial up to normalization, so polarization and diagonalization are inverse processes where normalizing constants are invertible.

 

 

 

 

 





## Boundary

Boundary

Requires that the polynomial be homogeneous of fixed degree and usually that factorials up to n be invertible in the base ring; does not directly apply to nonhomogeneous polynomials without decomposing into homogeneous components.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Related to but distinct from symmetrization: polarization produces a symmetric multilinear form whose diagonal yields the polynomial, whereas symmetrization acts on existing multilinear forms; confusion arises when switching between diagonal evaluation and averaging identities.

 

 

 

 

 





## Synthesis

Synthesis

Polarization is the normalization and finite-difference procedure that converts a homogeneous polynomial into its associated symmetric multilinear form and back by diagonal evaluation, making polynomial coefficients accessible in multilinear algebra and invariant theory.