15.6.4 Tensor Symmetric Multilinear Polarization Context

Tensor Symmetric Multilinear Polarization Context is the setting in which the generalized polarization identity is applied to recover a totally symmetric multilinear form of rank n from the single homogeneous degree-n polynomial obtained by evaluating that form on n copies of one vector. This context specifies the precise conditions, the exact combinatorial formula, and the underlying field requirements needed for polarization to succeed at rank n, extending the rank-2 relation between a symmetric bilinear form and its quadratic form to arbitrary rank.

Polarization in this general context is what makes the homogeneous polynomial Q(v) = B(v, ..., v) a faithful encoding of the full multilinear form B, rather than a lossy summary; without the polarization identity, it would not be clear that the seemingly simpler polynomial data determines the richer multilinear data uniquely. The context also clarifies the limits of polarization, particularly the dependence on being able to divide by factorial quantities that grow with the rank n.

The Generalized Polarization Formula

Statement of the Identity

For a totally symmetric multilinear form B of rank n and vectors v_1 through v_n, the polarization identity expresses B in terms of the homogeneous polynomial Q by an inclusion-exclusion sum over subsets of the arguments:

where the sum runs over every subset S of the argument labels, and the sign alternates according to how many labels are excluded from S.

Rank-2 Case as a Check

Restricting this general formula to n equal to two reproduces the earlier bilinear polarization identity, since the subsets of a two-element set are the empty set, the two singletons, and the full set, and combining their contributions with the prescribed alternating signs yields exactly Q(u+v) - Q(u) - Q(v), divided by 2! which equals 2.

Field Requirements for the Context

Division by Factorial Quantities

Because the polarization identity divides by n factorial, the context in which it applies is restricted to fields, such as the real or complex numbers, in which n factorial is invertible. Over a field of finite characteristic p, the identity fails to make sense once n reaches or exceeds p, since n factorial becomes zero or non-invertible in that field.

Rational and Real Coefficient Settings

In the ordinary rational, real, or complex number settings most commonly encountered when working with symmetric tensors, division by n factorial poses no obstruction for any finite rank n, so the polarization context is unrestricted in these settings and the identity holds for every rank.

Uniqueness and Faithfulness

The Polynomial Determines the Form

Within a valid polarization context, two totally symmetric multilinear forms that produce the same homogeneous degree-n polynomial Q must be identical as multilinear forms, since the polarization formula reconstructs B entirely from Q with no remaining ambiguity. This establishes a bijection between totally symmetric multilinear forms of rank n and homogeneous degree-n polynomials, mirroring the bijection already established at rank two between symmetric bilinear forms and quadratic forms.

Consistency Check via Diagonal Evaluation

Applying the polarization formula with all n vectors set equal to a single vector v should recover Q(v) itself; verifying this consistency confirms that the alternating sum correctly collapses back to the original diagonal evaluation, providing a sanity check on the identity's correctness within the given context.

Practical Role of the Polarization Context

Working With Polynomials Instead of Tensors

The polarization context allows problems naturally phrased in terms of homogeneous polynomials, such as questions from classical invariant theory or algebraic geometry, to be translated into equivalent problems about totally symmetric tensors, and vice versa, since the two descriptions are interchangeable whenever the polarization context's field requirements are satisfied.

Limits Outside the Context

When the field requirements fail, such as in small positive characteristic, a homogeneous polynomial may no longer determine a unique symmetric multilinear form, and the theory of divided powers is typically introduced as a substitute framework; this boundary marks the edge of the polarization context described here, which is confined to fields where the relevant factorials are invertible.