Can you write $1 - xyz$ in the form $p + q (1 - x^{2}-y^{2}-z^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z]$?
For instance, in the two variable case, $1 - xy = \frac{1}{2} + \frac{1}{2}(x-y)^{2} + \frac{1}{2}(1-x^{2}- y^{2})$. In this example, $q = \frac{1}{2}$, so we have a very simple expression. I'm looking for an analogous expression in three variables ($p$ and $q$ here can be anything as long as they are sums of squares). Also can we say anything in general for $n$ variables?
The degree of a combination of squares will always be an even number, but your initial polynomial has degree 3, so it won't be possible to write it in that way.