Polarization of an algebraic form

The technique

The fundamental ideas are as follows. Let f(\mathbf{u}) be a polynomial in n variables \mathbf{u} = \left(u_1, u_2, \ldots, u_n\right). Suppose that f is homogeneous of degree d, which means that f(t \mathbf{u}) = t^d f(\mathbf{u}) \quad \text{ for all } t.

Let \mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)} be a collection of indeterminates with \mathbf{u}^{(i)} = \left(u^{(i)}_1, u^{(i)}_2, \ldots, u^{(i)}_n\right), so that there are d n variables altogether. The polar form of f is a polynomial F\left(\mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)}\right) which is linear separately in each \mathbf{u}^{(i)} (that is, F is multilinear), symmetric in the \mathbf{u}^{(i)}, and such that F\left(\mathbf{u}, \mathbf{u}, \ldots, \mathbf{u}\right) = f(\mathbf{u}).

The polar form of f is given by the following construction F\left({\mathbf u}^{(1)}, \dots, {\mathbf u}^{(d)}\right) = \frac{1}{d!}\frac{\partial}{\partial\lambda_1} \dots \frac{\partial}{\partial\lambda_d}f(\lambda_1{\mathbf u}^{(1)} + \dots + \lambda_d{\mathbf u}^{(d)})|_{\lambda=0}. In other words, F is a constant multiple of the coefficient of \lambda_1 \lambda_2 \ldots \lambda_d in the expansion of f\left(\lambda_1 \mathbf{u}^{(1)} + \cdots + \lambda_d \mathbf{u}^{(d)}\right).

Examples

A quadratic example. Suppose that \mathbf{x} = (x,y) and f(\mathbf{x}) is the quadratic form f(\mathbf{x}) = x^2 + 3 x y + 2 y^2. Then the polarization of f is a function in \mathbf{x}^{(1)} = (x^{(1)}, y^{(1)}) and \mathbf{x}^{(2)} = (x^{(2)}, y^{(2)}) given by F\left(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right) = x^{(1)} x^{(2)} + \frac{3}{2} x^{(2)} y^{(1)} + \frac{3}{2} x^{(1)} y^{(2)} + 2 y^{(1)} y^{(2)}. More generally, if f is any quadratic form then the polarization of f agrees with the conclusion of the polarization identity.

A cubic example. Let f(x,y) = x^3 + 2xy^2. Then the polarization of f is given by F\left(x^{(1)}, y^{(1)}, x^{(2)}, y^{(2)}, x^{(3)}, y^{(3)}\right) = x^{(1)} x^{(2)} x^{(3)} + \frac{2}{3} x^{(1)} y^{(2)} y^{(3)} + \frac{2}{3} x^{(3)} y^{(1)} y^{(2)} + \frac{2}{3} x^{(2)} y^{(3)} y^{(1)}.

Mathematical details and consequences

The polarization of a homogeneous polynomial of degree d is valid over any commutative ring in which d! is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than d.

The polarization isomorphism (by degree)

For simplicity, let k be a field of characteristic zero and let A = k[\mathbf{x}] be the polynomial ring in n variables over k. Then A is graded by degree, so that A = \bigoplus_d A_d. The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree A_d \cong \operatorname{Sym}^d k^n where \operatorname{Sym}^d is the d-th symmetric power.

These isomorphisms can be expressed independently of a basis as follows. If V is a finite-dimensional vector space and A is the ring of k-valued polynomial functions on V graded by homogeneous degree, then polarization yields an isomorphism A_d \cong \operatorname{Sym}^d V^*.

The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on A, so that A \cong \operatorname{Sym}^{\bullet} V^* where \operatorname{Sym}^{\bullet} V^* is the full symmetric algebra over V^*.

Remarks

• For fields of positive characteristic p, the foregoing isomorphisms apply if the graded algebras are truncated at degree p - 1.
• There do exist generalizations when V is an infinite-dimensional topological vector space.

References

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, .