A Positivstellensatz which Preserves the Coupling Pattern of Variables

We specialize Schm\"udgen's Positivstellensatz and its Putinar and Jacobi and Prestel refinement, to the case of a polynomial $f\in R[X,Y]+R[Y,Z]$, positive on a compact basic semi algebraic set $K$ described by polynomials in $R[X,Y]$ and $R[Y,Z]$ only, or in $R[X]$ and $R[Y,Z]$ only (i.e. $K$ is a cartesian product). In particular, we show that the preordering $P(g,h)$ (resp. quadratic module $Q(g,h)$) generated by the polynomials $\{g_j\}\subset R[X,Y]$ and $\{h_k\}\subset R[Y,Z]$ that describe $K$, is replaced with $P(g)+P(h)$ (resp. $Q(g)+Q(h)$), so that the absence of coupling between $X$ and $Z$ is also preserved in the representation. A similar result applies with Krivine's Positivstellensatz involving the cone generated by $\{g_j,h_k\}$.