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Polya's Theorem with Zeros

Castle, Mariangely Fernandez (2008)
Dissertation (38 pages)
Committee Chair / Thesis Adviser: Powers, Victoria
Committee Members: Raman, Parimala
Research Fields: Mathematics
Keywords: Polynomials; Real Algebraic Geometry; Polya's Theorem
Program: Laney Graduate School, Math and Computer Science
Permanent url:


Polya's Theorem says that if a form (homogeneous polynomial) p in R[X] is positive on the standard simplex, then for sufficiently large N, the coefficients of (X1+· · ·+Xn)Np are positive. In 2001, Powers and Reznick established an explicit bound for the N in Polya's Theorem. The bound depends only on information about p, namely the degree and the size of the coefficients of p, and the minimum value of p on the simplex. This thesis is part of an ongoing project, started by Powers and Reznick in 2006, to understand exactly when Polya's Theorem holds if the condition "positive on n" is relaxed to "nonnegative on n", and to give bounds in this case. In this thesis, we will show that if a form p satisfies a relaxed version of Polya's Theorem, then the set of zeros of p is a union of faces of the simplex. We characterize forms which satisfy a relaxed version of Polya's Theorem and have zeros on vertices. Finally, we give a sufficient condition for forms with zero set a union of two-dimensional faces of the simplex to satisfy a relaxed version of Polya's Theorem, with a bound.

Table of Contents

1 Introduction 8
2 Preliminaries 13
3 A localized Polya's Theorem 18
4 Polya's Theorem With Zeros on Vertices 24
5 Zeros On A Two Dimensional Face 31


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