- Polarization of an algebraic form
In

mathematics , in particular inalgebra ,**polarization**is a technique for expressing ahomogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces amultilinear 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 , andrepresentation theory . Polarization and related techniques form the foundations forWeyl's invariant theory .**The technique**The fundamental ideas are as follows. Let "f"(

**u**) be a polynomial of "n" variables**u**= ("u"_{1}, "u"_{2}, ..., "u"_{n}). Suppose that "f" is homogeneous of degree "d", which means that:"f"("t"**u**) = "t"^{"d"}"f"(**u**) for all "t".Let

**u**^{(1)},**u**^{(2)}, ...,**u**^{(d)}be a collection ofindeterminate s with**u**^{(i)}= ("u"_{1}^{(i)}, "u"_{2}^{(i)}, ..., "u"_{n}^{(i)}), so that there are "dn" variables altogether. The**polar form**of "f" is a polynomial:"F"(**u**^{(1)},**u**^{(2)}, ...,**u**^{(d)})which is linear separately in each**u**^{(i)}(i.e., "F" is multilinear), symmetric among the**u**^{(i)}, and such that:"F"(**u**,**u**, ...,**u**)="f"(**u**).The polar form of "f" is given by the following construction:$F(\{old\; u\}^\{(1)\},dots,\{old\; u\}^\{(d)\})=frac\{1\}\{d!\}frac\{partial\}\{partiallambda\_1\}dotsfrac\{partial\}\{partiallambda\_d\}f(lambda\_1\{old\; u\}^\{(1)\}+dots+lambda\_d\{old\; u\}^\{(d)\}).$In other words, "F" is a constant multiple of the coefficient of λ

_{1}λ_{2}...λ_{d}in the expansion of "f"(λ_{1}**u**^{(1)}+ ... + λ_{d}**u**^{(d)}).**Examples***Suppose that

**x**=("x","y") and "f"(**x**) is thequadratic form :$f(\{old\; x\})\; =\; x^2\; +\; 3\; x\; y\; +\; 2\; y^2$.Then the polarization of "f" is a function in**x**^{(1)}= ("x"^{(1)}, "y"^{(1)}) and**x**^{(2)}= ("x"^{(2)}, "y"^{(2)}) given by:$F(\{old\; x\}^\{(1)\},\{old\; x\}^\{(2)\})\; =\; x^\{(1)\}x^\{(2)\}+frac\{3\}\{2\}x^\{(2)\}y^\{(1)\}+frac\{3\}\{2\}x^\{(1)\}y^\{(2)\}+2\; y^\{(1)\}y^\{(2)\}$*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}+ 2"xy"^{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 ofcharacteristic 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 thepolynomial ring in "n" variables over "k". Then "A" is graded by degree, so that:$A\; =\; igoplus\_d\; A\_d.$The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree:$A\_d\; cong\; Sym^d\; k^n$where "Sym"^{"d"}is the "d"-thsymmetric 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:$A\_d\; cong\; Sym^d\; V^*.$

**The algebraic isomorphism**Furthermore, the polarization is compatible with the algebraic structure on "A", so that:$A\; cong\; Sym^cdot\; V^*$where "Sym"

^{.}"V"^{*}is the fullsymmetric 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 dimensionaltopological vector space .

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