We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of the authors, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.2010 Mathematics Subject Classification. 26C05, 14M25, 32L15 .
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.2010 Mathematics Subject Classification. Primary 12D99; secondary 14P99, 26C99 .
Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D'Angelo, as well as that of a subsequent refinement by Varolin, involve subtle asymptotic estimates of the Bergman kernel. In this article, we give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.2010 Mathematics Subject Classification. 32L05, 32A26, 32H02 .
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