Polarization of an algebraic form

{{short description|Technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables}}

{{About|formulas for higher-degree polynomials|formula that relates norms to inner products|Polarization identity}}

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

$F\left(\mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)}\right)$

which is linear separately in each $\mathbf{u}^{(i)}$ (that is, $F$ is multilinear), symmetric in the $\mathbf{u}^{(i)},$ and such that

$F\left({\mathbf u}^{(1)}, \dots, {\mathbf u}^{(d)}\right) = \frac{1}{d!}\frac{\partial}{\partial\lambda_1} \dots \frac{\partial}{\partial\lambda_d}f(\lambda_1{\mathbf u}^{(1)} + \dots + \lambda_d{\mathbf u}^{(d)})|_{\lambda=0}.$

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, {{isbn|9780387260402}} .

Category:Abstract algebra

Category:Homogeneous polynomials