A Positivstellensatz which Preserves the Coupling Pattern of Variables

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We specialize Schmüdgen's Positivstellensatz and its Putinar and Jacobi and Prestel refinement, to the case of a polynomial f∈ 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 {gⱼ}⊂ R[X,Y] and {hₖ}⊂ 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 {gⱼ,hₖ}.