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Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...

Consider the following system of $n$ equations: \begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i \tag{$\star$} \end{equation} where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...