Tutte seminar - Victoria Powers

Pólya's Theorem with Zeros

Abstract:

Let R[X] := R[X₁,…,X_n]. Pólya's Theorem says that if a form (homogeneous polynomial) p\in R[X] is positive on the standard n-simplex Δ_n, then for sufficiently large N all the coefficients of (X₁+...+ X_n)^Np are positive. In 2001, the speaker and B. Reznick gave a bound on the N needed, in terms of the degree of p, the coefficients, and the minimum of p on Δ_n. This quantitative Pólya's Theorem has been used in many applications, in both pure and applied mathematics. This talk concerns work, jointly with M. Castle and B. Reznick, which is the culmination of a project to understand when Pólya's Theorem holds for forms if the condition "positive on Δ_n" is relaxed to "nonnegative on Δ_n". Using a certain "localized" Pólya's Theorem we are able to completely characterize such forms and to give a bound on the N needed.