A sparse version of Reznick's Positivstellensatz

Understand

If $f$ is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift.

• 220 (1995), pp.
• 75--97] states that there exists $k\in\mathbf{N}$ such that ${\| x \|^{2k}_2}f$ is a sum of squares of polynomials.
• Assuming that $f$ can be written as a sum of forms $\sum_{l=1}^p f_l$, where each $f_l$ depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz.