Technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables
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
be a polynomial in
variables
Suppose that
is homogeneous of degree
which means that
Let
be a collection of indeterminates with
so that there are
variables altogether. The polar form of
is a polynomial
which is linear separately in each
(that is,
is multilinear), symmetric in the
and such that
The polar form of
is given by the following construction
In other words,
is a constant multiple of the coefficient of
in the expansion of
Examples
A quadratic example. Suppose that
and
is the quadratic form
Then the polarization of
is a function in
and
given by
More generally, if
is any quadratic form then the polarization of
agrees with the conclusion of the polarization identity.
A cubic example. Let
Then the polarization of
is given by
Mathematical details and consequences
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)
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
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
where
is the
-th symmetric power of the
-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
The algebraic isomorphism
Furthermore, the polarization is compatible with the algebraic structure on
, so that
where
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
- Homogeneous function – Function with a multiplicative scaling behaviour
References
- Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402 .