Polarization of An Algebraic Form - The Technique

The Technique

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

f(t u) = td f(u) for all t.

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

F(u(1), u(2), ..., u(d))

which is linear separately in each u(i) (i.e., F is multilinear), symmetric in the u(i), and such that

F(u,u, ..., u)=f(u).

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)).

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