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 fundamental ideas are as follows.
is homogeneous of degree
be a collection of indeterminates with
variables altogether.
The polar form of
which is linear separately in each
is multilinear), symmetric in the
The polar form of
is a constant multiple of the coefficient of
is the quadratic form
is any quadratic form then the polarization of
agrees with the conclusion of the polarization identity.
The polarization of a homogeneous polynomial of degree
is valid over any commutative ring in which
In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than
be a field of characteristic zero and let
be the polynomial ring in
is graded by degree, so that
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
-th symmetric power.
These isomorphisms can be expressed independently of a basis as follows.
is a finite-dimensional vector space and
-valued polynomial functions on
graded by homogeneous degree, then polarization yields an isomorphism
Furthermore, the polarization is compatible with the algebraic structure on
is the full symmetric algebra over