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## Summary

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 $\mathbf {x} ^{(1)}=\left(x^{(1)},y^{(1)}\right)$  and $\mathbf {x} ^{(2)}=\left(x^{(2)},y^{(2)}\right)$  given by
$F\left(\mathbf {x} ^{(1)},\mathbf {x} ^{(2)}\right)=x^{(1)}x^{(2)}+{\frac {3}{2}}x^{(2)}y^{(1)}+{\frac {3}{2}}x^{(1)}y^{(2)}+2y^{(1)}y^{(2)}.$

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 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

$A\cong \operatorname {Sym} ^{\cdot }V^{*}$

where $\operatorname {Sym} ^{\cdot }V^{*}$  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.