Vol. 132, No. 1, 1988

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Representing polynomials by positive linear functions on compact convex polyhedra

David E. Handelman

Vol. 132 (1988), No. 1, 35–62

If K is a compact polyhedron in Euclidean d-space, defined by linear inequalities, βi 0, and if f is a polynomial in d variables that is strictly positive on K, then f can be expressed as a positive linear combination of products of members of {βi}. 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 {βi}.

Mathematical Subject Classification 2000
Primary: 52A25, 52A25
Secondary: 13F99, 19E99, 19K14, 46L99
Received: 16 May 1985
Revised: 26 February 1987
Published: 1 March 1988
David E. Handelman