Technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables
This article is about formulas for higher-degree polynomials. For formula that relates norms to inner products, see
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[edit]
The fundamental ideas are as follows. Let
be a polynomial in
variables
Suppose that
is homogeneous of degree
which means that
![{\displaystyle f(t\mathbf {u} )=t^{d}f(\mathbf {u} )\quad {\text{ for all }}t.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff945e5de99eb013e059a956dc202a4ff92ccda)
Let
be a collection of indeterminates with
so that there are
variables altogether. The polar form of
is a polynomial
![{\displaystyle F\left(\mathbf {u} ^{(1)},\mathbf {u} ^{(2)},\ldots ,\mathbf {u} ^{(d)}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ef1c21399ffc4f9d192c30dcac4e189e9ad8a4)
which is linear separately in each
![{\displaystyle \mathbf {u} ^{(i)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/886340929b3f17ed2f8df44eb12f536bce14f720)
(that is,
![{\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
is multilinear), symmetric in the
![{\displaystyle \mathbf {u} ^{(i)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8513221e735be87d0e8b539be5171dba59d1a960)
and such that
![{\displaystyle F\left(\mathbf {u} ,\mathbf {u} ,\ldots ,\mathbf {u} \right)=f(\mathbf {u} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f078f140efd291a0d14ef194b130598387f8791a)
The polar form of
is given by the following construction
![{\displaystyle 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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6493fa2f541cf8e026022f92a49a4d47145f7cdd)
In other words,
![{\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
is a constant multiple of the coefficient of
![{\displaystyle \lambda _{1}\lambda _{2}\ldots \lambda _{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bda2f7eaf6de48efed5e36c36bd80d2fa24cf63)
in the expansion of
Examples[edit]
A quadratic example. Suppose that
and
is the quadratic form
![{\displaystyle f(\mathbf {x} )=x^{2}+3xy+2y^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fff9329e360714577a439bc5e988fe1d8beff886)
Then the polarization of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is a function in
![{\displaystyle \mathbf {x} ^{(1)}=\left(x^{(1)},y^{(1)}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e59f7f2c3cc25a8cf77be775e5f797653cc672)
and
![{\displaystyle \mathbf {x} ^{(2)}=\left(x^{(2)},y^{(2)}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/644c855a03d4cf1fccf52dff7a2a55e594b17dbf)
given by
![{\displaystyle 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)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ae919e81a35d24e03f83cc9e5e62198572924e)
More generally, if
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is any quadratic form then the polarization of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
agrees with the conclusion of the
polarization identity.
A cubic example. Let
Then the polarization of
is given by
![{\displaystyle 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)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a99ab76caf96335daa9388ca988f0539de3cf8)
Mathematical details and consequences[edit]
The polarization of a homogeneous polynomial of degree
is valid over any commutative ring in which
is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than
The polarization isomorphism (by degree)[edit]
For simplicity, let
be a field of characteristic zero and let
be the polynomial ring in
variables over
Then
is graded by degree, so that
![{\displaystyle A=\bigoplus _{d}A_{d}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68390dc3ee5344698c9de0dbd7603334c00a521)
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
![{\displaystyle A_{d}\cong \operatorname {Sym} ^{d}k^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a43c70f040e36bf39a6290bd155d98da731763f1)
where
![{\displaystyle \operatorname {Sym} ^{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/422ebbccd2346b60adcb5728a803133bee9f6571)
is the
![{\displaystyle d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
-th
symmetric power of the
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-dimensional space
These isomorphisms can be expressed independently of a basis as follows. If
is a finite-dimensional vector space and
is the ring of
-valued polynomial functions on
graded by homogeneous degree, then polarization yields an isomorphism
![{\displaystyle A_{d}\cong \operatorname {Sym} ^{d}V^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce1bd3015ad05850c2f25c1e45c33aca8ea07b0)
The algebraic isomorphism[edit]
Furthermore, the polarization is compatible with the algebraic structure on
, so that
![{\displaystyle A\cong \operatorname {Sym} ^{\bullet }V^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d36cca7fae2161f763e9d50901f7c5cbae90a71)
where
![{\displaystyle \operatorname {Sym} ^{\bullet }V^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e438a3b06df9e146866100f4036aff98f727ab14)
is the full
symmetric algebra over
- For fields of positive characteristic
the foregoing isomorphisms apply if the graded algebras are truncated at degree ![{\displaystyle p-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/695dddf8d8bd947194d5e41447b0228b298deb30)
- There do exist generalizations when
is an infinite dimensional topological vector space.
See also[edit]
References[edit]