The Aronhold Method, Polarization
2.1 Polarizations
Before proceeding, let us recall, in a language suitable for our purposes, the usual Taylor-Maclaurin expansion. Consider a function \( F(x)\) of a vector variable \( x \in V \). Under various types of assumptions we have a development for the function \( F(x+y) \) of two vector variables. For our purposes, we may restrict our considerations to polynomials and develop
$$F(x + y) := \sum^{i = 0}_\infty F_i(x,y), $$ where by definition \( F_i (x, y) \) is homogeneous of degree \( i \) in \( y \) (of course for polynomials the sum is really finite). herefore, for any value of a parameter \( \lambda \), we have \( F(x + \lambda y) := \sum^{i = 0}_\infty \lambda^i F_i(x,y) \). If F is also homogeneous of degree \( k \) we have
$$ \begin{align}
\sum^{i = 0}_\infty \lambda^k F_i(x,y) &= \lambda^k F(x+y) = F(\lambda (x + y)) = \\
& F(\lambda x + \lambda y) = \sum^\infty_{i=0} \lambda^i f_i (\lambda x, y)
\end{align} $$
and we deduce that \( F_i(x, y) \) is also homogeneous of degree \( k - i \) in \( x \).