Polarization of an algebraic form

In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.

Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory.

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 $$\mathbf{x}^{(1)} = \left(x^{(1)}, y^{(1)}\right)$$ and $$\mathbf{x}^{(2)} = \left(x^{(2)}, y^{(2)}\right)$$ 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 of the $$n$$-dimensional space $$k^n.$$

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.