As mentioned here on page 4, Wakeford proved that a generic algebraic form over $\mathbb{C}$
$$\sum_{i_1,i_2,...,i_d=1}^dA_{i_1i_2...i_d}x_{i_1}x_{i_2}...x_{i_d}$$
can always be transformed by an invertible linear change of variables over $\mathbb{C}$ so that the coeffcient of each $x_i^d$ is $1$ and the coefficient of each $x_i^{d-1}x_j$ is $0$. This is a sort of higher degree analog of finding an orthonormal basis for a quadratic form.
I was wondering to what extent this result survives over $\mathbb{R}$ (i.e. both the coefficients and the linear transformations are real). Obviously for even degree the coefficient of $x_i^d$ could at best be $\pm1$. Aside from that I can't see anything obviously wrong with an analogous result, but the standard techniques for canonical forms over $\mathbb{C}$ don't seem viable over $\mathbb{R}$. I'm primarily interested in the degree 4, positive-definite case if that helps any.