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Jun 22, 2016 at 11:55 comment added David Handelman Rings of this type, generalized to arbitrary integral polytopes and permitting other coefficients than 1, are considered (at length) in Positive polynomials and product type actions of compact groups (AMS Memoirs, 320, 1985) and especially Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem (SLN 1282, 1987), both by me (there are also later publications dealing with them). But they don't address your specific problem.
May 3, 2013 at 13:30 history asked Neil Strickland CC BY-SA 3.0