Pólya’s Theorem with zeros

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Journal of Symbolic Computation, ISSN: 0747-7171, Vol: 46, Issue: 9, Page: 1039-1048

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Castle, Mari; Powers, Victoria; Reznick, Bruce
Elsevier BV
Mathematics; POLYNOMIAL rings; COMBINATORIAL analysis; VARIABLES; NON-negative matrices; MATHEMATICAL analysis; MATHEMATICAL constants; Analysis; Applied Mathematics
article description
Let R[X] be the real polynomial ring in n variables. Pólya’s Theorem says that if a homogeneous polynomial p∈R[X] is positive on the standard n -simplex Δn, then for sufficiently large N all the coefficients of (X1+⋯+Xn)Np are positive. We give a complete characterization of forms, possibly with zeros on Δn, for which there exists N so that all coefficients of (X1+⋯+Xn)Np have only nonnegative coefficients, along with a bound on the N needed.