Resumen de A quantitative Pólya's Theorem with zeros

Mari Castle, Victoria Powers, Bruce Reznick

• Let . Pólya's Theorem says that if a form (homogeneous polynomial) is positive on the standard n-simplex D-n, then for sufficiently large N all the coefficients of (X1++Xn)Np are positive. The work in this paper is part of an ongoing project aiming to explain when Pólya's Theorem holds for forms if the condition "positive on Dn" is relaxed to "nonnegative on Dn", and to give bounds on N. Schweighofer gave a condition which implies the conclusion of Pólya's Theorem for polynomials . We give a quantitative version of this result and use it to settle the case where a form is positive on n, apart from possibly having zeros at the corners of the simplex.