Polarization of an algebraic form

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 multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal. 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 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 f(u) be a polynomial in n variables u = (u1, u2, ..., un). Suppose that f is homogeneous of degree d, which means that Let u(1), u(2), ..., u(d) be a collection of indeterminates with u(i) = (u1(i), u2(i), ..., un(i)), so that there are dn variables altogether. The polar form of f is a polynomial which is linear separately in each u(i) (i.e., F is multilinear), symmetric in the u(i), and such that The polar form of f is given by the following construction In other words, F is a constant multiple of the coefficient of λ1 λ2...λd in the expansion of f(λ1u(1) + ... + λdu(d)). Then the polarization of f is a function in x(1) = (x(1), y(1)) and x(2) = (x(2), y(2)) given by The polarization of a homogeneous polynomial of degree d is valid over any commutative ring in which d! is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than d.