I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits_{k=1}^n x_k^4, x \in \mathbb{R^n}$ as a sum of squares of real quadratic forms. Murray Marshall in his book "Positive polynomials and sums of squares" show that quartic diagonal is in the interior of the cone of sum of squares. Does anyone knows details about such representation.

In particular, suppose $\sum \limits_{k=1}^n x_k^4= \sum_p (x^T A_p x)^2$ ($x$ is a column vector, and $x^T$ its transpose, $A_p \in \mathbb{R^n \times R^n}$) then what $Q\in \mathbb{R^n \times R^n}$ can be represented by the expression $\sum_p (x^T A_p x - {x^*}^T A_p x^*)^2= \sum \limits_{k=1}^n x_k^4 - 2x^T Q x+ const$, $x^*$ is a point in $\mathbb{R^n}$. More specifically, whether $Q$ is dense around identity matrix (in a small neighborhood).