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 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
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