Polarization of an algebraic form

Template:Short description Template:About 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 $f(𝐮)$ be a polynomial in $n$ variables $𝐮=(u_1,u_2,\dots ,u_n).$ Suppose that $f$ is homogeneous of degree $d,$ which means that $$f(t𝐮)=t^{d}f(𝐮)\text{ for all }t.$$

Let ${𝐮}^{(1)},{𝐮}^{(2)},\dots ,{𝐮}^{(d)}$ be a collection of indeterminates with ${𝐮}^{(i)}=(u_1^{(i)},u_2^{(i)},\dots ,u_n^{(i)}),$ so that there are $dn$ variables altogether. The polar form of $f$ is a polynomial $$F({𝐮}^{(1)},{𝐮}^{(2)},\dots ,{𝐮}^{(d)})$$ which is linear separately in each ${𝐮}^{(i)}$ (that is, $F$ is multilinear), symmetric in the ${𝐮}^{(i)},$ and such that $$F(𝐮,𝐮,\dots ,𝐮)=f(𝐮).$$

The polar form of $f$ is given by the following construction $$F({𝐮}^{(1)},\dots ,{𝐮}^{(d)})=\frac{1}{d!}\frac{\partial }{\partial {\lambda }_1}\dots \frac{\partial }{\partial {\lambda }_d}f({\lambda }_1{𝐮}^{(1)}+\dots +{\lambda }_d{𝐮}^{(d)}){|}_{\lambda =0}.$$ In other words, $F$ is a constant multiple of the coefficient of ${\lambda }_1{\lambda }_2\dots {\lambda }_d$ in the expansion of $f({\lambda }_1{𝐮}^{(1)}+\cdots +{\lambda }_d{𝐮}^{(d)}).$

Examples

A quadratic example. Suppose that $𝐱=(x,y)$ and $f(𝐱)$ is the quadratic form $$f(𝐱)=x^2+3xy+2y^2.$$ Then the polarization of $f$ is a function in ${𝐱}^{(1)}=(x^{(1)},y^{(1)})$ and ${𝐱}^{(2)}=(x^{(2)},y^{(2)})$ given by $$F({𝐱}^{(1)},{𝐱}^{(2)})=x^{(1)}{x}^{(2)}+\frac{3}{2}{x}^{(2)}{y}^{(1)}+\frac{3}{2}{x}^{(1)}{y}^{(2)}+2y^{(1)}{y}^{(2)}.$$ 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 $f(x,y)=x^3+2x{y}^2.$ Then the polarization of $f$ is given by $$F(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)})=x^{(1)}{x}^{(2)}{x}^{(3)}+\frac{2}{3}{x}^{(1)}{y}^{(2)}{y}^{(3)}+\frac{2}{3}{x}^{(3)}{y}^{(1)}{y}^{(2)}+\frac{2}{3}{x}^{(2)}{y}^{(3)}{y}^{(1)}.$$

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 $A=k[𝐱]$ be the polynomial ring in $n$ variables over $k.$ Then $A$ is graded by degree, so that $$A=\underset{d}{⨁}{A}_d.$$ The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree $$A_d\cong {\mathrm{Sym}}^d{k}^n$$ where ${\mathrm{Sym}}^d$ 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 $$A_d\cong {\mathrm{Sym}}^d{V}^{*}.$$

The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on $A$, so that $$A\cong {\mathrm{Sym}}^{\bullet }{V}^{*}$$ where ${\mathrm{Sym}}^{\bullet }{V}^{*}$ 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.

See also

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References

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