# 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 $\mathbf {u} ^{(i)}=\left(u_{1}^{(i)},u_{2}^{(i)},\ldots ,u_{n}^{(i)}\right),$ so that there are $dn$ 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 $f(\mathbf {x} )=x^{2}+3xy+2y^{2}.$ 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 $F\left(\mathbf {x} ^{(1)},\mathbf {x} ^{(2)}\right)=x^{(1)}x^{(2)}+{\frac {3}{2}}x^{(2)}y^{(1)}+{\frac {3}{2}}x^{(1)}y^{(2)}+2y^{(1)}y^{(2)}.$ 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.