#
Polya's Theorem with Zeros

Castle, Mariangely Fernandez
(2008)
Polya's Theorem with Zeros

**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:**http://pid.emory.edu/ark:/25593/15xr6

## Abstract

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 (X_{1}+· ·
·+X_{n})^{N}p 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

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