Representing polynomials by positive linear functions on compact convex polyhedra

If K is a compact polyhedron in Euclidean έ/-space, defined by linear inequalities, βt> 0, and if / is a polynomial in d variables that is strictly positive on AT, then / can be expressed as a positive linear combination of products of members of {/?,}. In proving this and subsidiary results, we construct an ordered ring that is a complete AGL(d, R)-invariant for K, and discuss some of its properties. For example, the ordered ring associated to K admits the Riesz interpo-lation property if and only if it is AGL(d, R)-equivalent to a product of simplices. This is exploited to show that certain polynomials are not in the positive cone generated by the set {/?,}. Let L be a subfield of the real numbers, and let j8, = y ai4+ \\ (Ϊ = 1,2, 3,..., J) be linear polynomials ("linear forms") in the d variables {Xj}9 with coefficients from L. Suppose the convex poly-hedron in R ^ defined by K = f){βi)~~ι{[0,oo)) is compact and has interior. Let / be a polynomial in the d variables with entries from L, such that the restriction, f\, is strictly positive. Then our first result (1.3) asserts that / may be represented as a combination with coefficients from L n R+ (that is, positive numbers in L) of terms that are products of the original set of /Γs that determine K. If / van-ishes at only a vertex of K (and is strictly positive elsewhere), this decomposition does not hold in general (§ΠI). Our second principal result concerns the Riesz decomposition prop-erty in an ordered ring naturally associated to K, and leads to some in-teresting geometric characterizations of those polytopes that are affine-ly homeomorphic to products of simplices. With K defined as above, define a monomial (in the βfs) to be a polynomial in the JΓs that can be expressed as a product of the form βw βw(\\)βw(2) m m, βw(s) where w(k) is a non-negative integer, and w is the s-tuple (w(l), w(2),...,w(s)). Define RL[K] (or simplv R[K] if there is no ambiguity about the coefficient field L) to be the polynomial ring,