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- W2037009101 startingPage "35" @default.
- W2037009101 abstract "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 interpolation 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 polyhedron 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, fK, 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 / vanishes 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 property in an ordered ring naturally associated to K, and leads to some interesting geometric characterizations of those polytopes that are affinely 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)" @default.
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- W2037009101 date "1988-03-01" @default.
- W2037009101 modified "2023-10-01" @default.
- W2037009101 title "Representing polynomials by positive linear functions on compact convex polyhedra" @default.
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- W2037009101 doi "https://doi.org/10.2140/pjm.1988.132.35" @default.
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