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.
Inmathematics, 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.
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 formof 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
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
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 fieldofcharacteristic zero or whose characteristic is strictly greater than
For simplicity, let be a field of characteristic zero and let be the polynomial ringin variables over Then isgradedbydegree, 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