A Positivstellensatz for forms on the positive orthant

摘要

Let p be a nonconstant form in R[x(1),..., x(n)] with p(1,..., 1) > 0. If p(m) has strictly positive coefficients for some integer m >= 1, we show that p(m) has strictly positive coefficients for all sufficiently large m. More generally, for any such p and any form q that is strictly positive on (R+)(n){0}, we show that the form p(m) q has strictly positive coefficients for all sufficiently large m. This result can be considered as a strict Positivstellensatz for forms relative to (R+)(n){0}. We give two proofs, one based on results of Handelman, the other on techniques from real algebra.